STRUCTURAL  DETAILS 


OF 


HIP  AND  VALLEY  RAFTERS 


BY 

CARLTOX   THOMAS   BISHOP,  C.E. 

ASSISTANT  PROFESSOR  OF  CIVIL  KN' .  I  \\:i  KINi ;.  sil  II'KIKI.I)  SCIENTIFIC  SCHOOL  OF  YAL.K  UNIVERSITY 

M  l:l.\    m:\Kis\I\N   FOR  THE  AMERICAN  BRUME  COMPANY  AND 
CHIEF  DRAFTSMAN  FOR  THE  HAY  FOUNDRY  AND  IRON  WORKS 


FIRST   EDITION 

FIRST    THOUSAND 


M.W     YORK 

JOHN  WILEY  AND  SONS 

LONDON:    CHAPMAN    AND    HALL,  LIMITED 
1912 


COPYRIGHT,  1912, 

BY 
CARLTON  T.   BISHOP 


Stanhope  iprcss 

F.    H.GILSON   COMPANY 
BOSTON,  U.S.A. 


PREFACE 


Tins  hook  is  written  to  meet  the  requirements  of  structural  draftsmen 
when  solving  problems  in  hip  an<l  valley  eoiistniction.  No  attempt  has 
IN  en  made  to  show  the  application  to  skew  portals,  hoppers,  or  chutes, 
but,  in  view  of  the  similarity  of  the  angles,  it  is  felt  that  the  formulas 
given  will  lx>  of  great  assistance  to  the  draftsmen  when  dealing  with  such 
problems.  Complete  directions  arc  given  for  making  the  shop  drawings 
for  the  steel  work  of  intersecting  roofs  and  similar  structures.  The  notes 
for  the  various  cases  are  arranged  for  convenient  reference,  and  are  illus- 
trated by  general  drawings  and  typical  problems.  The  necessary  numeri- 
cal values  may  be  obtained  either  algebraically  or  graphically,  and  both 
methods  are  fully  explained.  Much  of  the  calculation  is  simplified  by 
means  of  tables,  which  give  values  for  use  in  the  most  common  cases. 

During  his  experience  as  structural  draftsman  the  author  either  made 
or  checked  many  drawings  of  hip  and  valley  rafters  and  drawings  of 
a  similar  nature,  using  numerous  systems  of  calculation,  both  published 
and  unpublished.  He  found,  however,  that  no  system  fulfilled  his  needs 
without  alteration  or  addition,  and  that  while  the  majority  was  effective 

NEW  HAVEN,  CONN.,  September,  1912. 


in  showing  the  derivations  of  the  more  difficult  angles,  few  explained  the 
application  of  these  angles,  or  referred  to  the  additional  values  required. 
These  values  may  be  easily  found  by  an  experienced  man,  but  the  author 
has  observed  that  the  average  draftsman  will  waste  much  time  in  procuring 
them,  since  he  lacks  a  clear  understanding  of  the  requirements.  It  was  felt 
by  the  author  and  his  fellow  draftsmen  that  there  existed  a  demand  for 
a  complete  treatise  of  this  nature,  and  accordingly  this  book  has  been 
developed.  It  is  hoped  that  it  will  prove  of  service  to  the  experienced 
draftsman,  as  well  as  to  the  novice  who  attempts  to  solve  a  problem  of 
this  character. 

The  author  desires  to  express  his  appreciation  of  the  encouragement  and 
helpful  criticisms  given  by  Prof.  J.  C.  Tracy,  of  Yale  University.  Grate- 
ful acknowledgment  is  also  given  to  Mr.  E.  R.  St.  John,  of  Pittsburgh,  for 
checking  the  more  important  drawings  and  formulas,  and  to  Messrs. 
L.  D.  Rights,  of  New  York,  and  W.  H.  McKinley,  of  Pitteburg,  for  their 
prepublication  criticisms. 

C.  T.  B. 


259667 

iii 


TABLE   OF  CONTENTS 


PREFACE 


CHAPTER  I 
GENERAL  OUTLINE 

Object 1 

Definitions 1 

Arrangement 2 

Working  Lines 2 

Column  Connections 2 

Computation 2 

Consistent  Accuracy 2 

Notation ,  3 


CHAPTER   II 
FLANGE   CONNECTION 


PAOE  PAGE 

.  iii      Explanatory  Notes  and  Suggestions 19 

Illustrative  Problem;  Case  I  (o);  Hip  Rafter 20 

Illustrative  Problem;  Case  I  (o);  Fig.  9 21 

Valley  Rafter 22 

Fig.  10 23 

Calculation 24 


I  (a);  Hip  Rafter;   Buildings  at  Right  Angle;   Unequal  Pitches. 

I  (a) 

I  (b);  Valley  Rafter;  Buildings  at  Right  Angle;  Unequal  Pitches. 

1(6) 

II  (a);  Hip  Rafter;  Buildings  at  Oblique  Angle;  Unequal  Pitches. 
II  (a) 


Formulas;  Case 
Fig.  2;  Case 
Formulas;  Case 
Fig.  3;  Case 
Formulas;  Case 
Fig.  4;  Case 
Formulas;  Case  II  (b) ;  Valley  Rafter;  Buildings  at  ObliqueAngle;  Unequal  Pitches. 

Fig.  5;       Case  II  (b) 

Formulas;  Case  III  (o);  Hip  Rafter;  Buildings  at  Right  Angle;  Equal  Pitches.     . 

Fig.  6;        Case  III  (a) 13 

Formulas;  Case  III  (6) ;   Valley  Rafter;  Buildings  at  Right  Angle;  Equal  Pitches.     14 

Fig.  7;        Case  111(6) 15 

Blank  Form  for  Calculation;  Case     I 16 

Blank  Form  for  Calculation;  Case    II 17 

Blank  Form  for  Calculation;  Case  III 18 


4 
5 
6 

7 
8 
9 

10 
11 

12 


Illustrative  Problem;  Case  I  (6); 
Illustrative  Problem;  Case  I  (6); 
Illustrative  Problem;  Case  I  (a); 


Illustrative  Problem;  Case  I  (6);  Calculation 25 

CHAPTER   III 
WEB  CONNECTION 

Formulas;  Case  IV  (a) ;  Hip  Rafter;   Buildings  at  Right  Angle;   Unequal  Pitches.     26 

Fig.  11;      Case  IV  (a) 27 

Formulas;  Case  IV  (6) ;  Valley  Rafter;  Buildings  at  Right  Angle;  Unequal  Pitches.     28 

Fig.  12;      Case  IV  (6) 29 

Formulas;  Case    V  (o);  Hip  Rafter;  Buildings  at  Oblique  Angle;  Unequal  Pitches.     30 

Fig.  13;      Case    V  (a) 

Formulas;  Case    V  (6) ;  Valley  Rafter;  Buildings  at  Oblique  Angle;  Unequal  Pitches. 
Fig.  14;      Case    V  (6) 


Formulas;  Case  VI  (a);  Hip  Rafter;  Buildings  at  Right  Angle;  Equal  Pitches. 


31 
32 
33 
34 


Fig.  15;      Case  VI  (a) 35 

Formulas;  Case  VI  (6);  Valley  Rafter;  Buildings  at  Right  Angle;   Equal  Pitches    .  36 

Fig.  16;      Case  VI  (b) 37 

Blank  Form  for  Calculation;  Case  IV 38 

Blank  Form  for  Calculation;  Case   V 39 

Blank  Form  for  Calculation ;  Case  VI 40 

Explanatory  Notes  and  Suggestions 41 


Illustrative  Problem ;  Case  IV  (a) ; 

Illustrative  Problem;  Case  IV  (a); 

Illustrative  Problem ;  Case  IV  (6) ; 

Illustrative  Problem;  Case  IV  (b); 

Illustrative  Problem;  Case  IV  (o); 

Illustrative  Problem;  Case  IV  (b); 


Hip  Rafter 42 

Fig.  17 43 

Valley  Rafter 44 

Fig.  18 45 

Calculation 46 

Calculation 47 


IV 


TABLE  OF  CONTENTS 


(  IIAITF.R    IV 

NOTES   ON   OTHER   CASES 

PAOI 

('Iniinel  Purlins  wli>                                 tin- Otlirr  \\'ny 48 

1-l.f.iiii  Purlins 48 

7,-l.ar  Purlins 48 

Purlins 48 

Purlins        48 

Purlin-;    Hip  Rafter 49 

Dallas;   Tee  Purlins;   Valley  Rafter 60 

P.I;     I'..-  Purlins    Hip  Rafter 51 

JO;  Tee  Purlins;   Valley  Rafter 51 

< -II  \ITKR  V 
DERIVATION   OF  FORMULAS 

Formula  (7) 52 

Formula  (9) " 52 

Formula  (13) 53 

52 

52 

nula  (18) 54 

Formula  (19) 56 

Formula  (21) 57 

Formula  (30) 58 

Formula  (31) 58 

Formula  (:>4) 59 

Formula  (55) 60 

Formula  i  :.7: 61 

Formula  (59) 61 

1  orniula  (84) .61 


CHAPTF.R   VI 
GRAPHIC   METHOD   OF  DETERMINING   ANGLES 


Fig.  30; 
FiK.  :tl; 
Fig.  32; 
Fig.  33; 
Fig.  34; 
FiK.  3.r, ; 
Fig.  30; 
Fig.  37; 
Fig.  38; 
Fig.  39; 
Fig.  40; 
Fig.  41; 
Fig.  42; 
Fig.  43; 


Flange  Connection;  Angles  W  and  W" 63 

Flange  Connection;  Angles  X'  and  X" 63 

Flange  Connection ;  Angles  Y'  and  Y" 63 

Flange  Connection;  Combined  Layout;  Cose  I 64 

Flange  Connection;  Combined  Layout;  Cose  II 64 

\\i-li('oniieetion;  Angles  W  and  W" 65 

Wcl)  Connection;  Angles  X'  and  A'" 65 

\\vii  Connection;  Angles  Y'  and  Y" 65 

Web  Connection;  Angles  Z' and  Z" 66 


\\diConnection;  Combined  Layout;  Cose  IV 66 

Web  Connection;  Combined  Layout;  Case  V 66 

Tee  Purlins;  Angles  X'  and  X" 67 

Tee  Purlins;  Combined  Layout;  Case  I 67 

Tee  Purlins;  Combined  Layout;  Case  II j.  67 

TAHI.KS 
VALUES  AND   LOGARITHMS  FOR  COMMON  CASES 

Pitches  J  and  30" 68 

Pitches  land  1 68 

Pitches  j  and  J 68 

Pitches  1  and  I 60 

Pitches  30°  and  J 69 

Pitches  30°  and  } 69 

Pitches  30°  and  J 70 

Pitches  J  and  J 70 

Pitches  J  and  J 70 

Pitches  J  and  i 71 

Equal  pitches  } 71 

Equal  pitches  30° 71 

Equal  pitches  \ 72 

Equal  pitches  \ 72 

Equal  pitches  } 72 


• 


HIP  AND  VALLEY   RAFTERS 


CHAPTER  I 
GENERAL  OUTLINE 


TIIK  structural  draftsman  will  occasionally,  if  not  frequently,  encounter 
a  building  which  contains  one  or  more  hip  or  valley  rafters.  Work  of  this 
character  involves  a  certain  amount  of  calculation  peculiar  to  itself,  and 
unle»  one  comes  into  contact  with  a  problem  of  this  kind  repeatedly,  he 
will  probably  become  so  much  out  of  practice  that  he  will  be  forced  to  spend 
considerable  time  in  review  before  he  can  ascertain  the  desired  results. 

Oliji  rl.  It  is  the  purjwse  of  this  book  to  present  the  subject  of  Hip  and 
Valley  construction  so  completely  that  anyone  with  a  reasonable  knowledge 
of  structural  details  and  of  trigonometry  can  make  working  drawings 
which  shall  give  all  necessary  information  to  the  shop  without  useless 
refinements.  The  work  is  made  definite  by  being  outlined  in  full,  and 
forms  of  tabulation  are  given  with  the  hope  that  valuable  time  may  be 
saved. 

The  most  common  types  of  hip  and  valley  construction  are  considered 
in  detail,  the  notes  being  arranged  for  convenient  reference.  These  types 
involve  channel  purlins  which  connect  either  to  the  flange  or  to  the  web  of 
the  hi]>  or  valley  rafter.  The  rafter  web  is  assumed  to  be  vertical  and  the 
flange  normal  to  the  web,  as,  for  example,  a  rolled  I-beam,  or  plate  and 
angles  built  in  the  form  of  an  I  or  a  T.  Suggestions  are  also  given  to  aid 
the  draftsman  in  modifying  these  types  to  fulfill  his  requirements  when  he 
meets  other  conditions. 

Definitions.  —  In  building  construction,  two  roofs  often  intersect  to 
form  either  a  hip  or  a  valley.  If  the  two  roofs  are  so  arranged  that  the 
drainage  is  away  from  the  line  of  intersection  a  "Hip  "  results,  but  if  the 
drainage  is  toward  the  intersection  a  "Valley"  is  formed.  (See  Fig.  1.) 
The  rafter  which  supports  the  purlin-  of  both  roofs  at  their  intersection  is, 
accordingly,  termed  "Hip  Rafter"  or  "Valley  Rafter." 


ELEVATIQS 
Fia.  1. 


•*: 


HIP  AND  VALLEY  RAFTERS 


Arrangement.  —  The  notes  are  classified  and  arranged  to  simplify  the 
reference  to  any  desired  section.  In  Chapter  II  are  discussed  the  connec- 
tions of  channel  purlins  to  the  flange  of  the  hip  or  valley  rafter,  giving  the 
ordinary,  the  general,  and  the  special  cases  of  each.  A  general  drawing, 
showing  all  values,  is  given  for  each  case  and  the  corresponding  formulas 
are  tabulated  on  the  opposite  page.  A  blank  form  is  given  for  each  case 
to  serve  as  a  guide  in  calculation  and  tabulation  and  to  indicate  the  loga- 
rithms to  be  used  in  later  computation,  thus  reducing  the  amount  of  labor 
involved.  A  few  explanatory  notes  follow,  with  suggestions  for  the  use  of 
the  values  obtained,  and,  finally,  two  illustrative  problems  are  given  in 
detail,  including  both  drawings  and  calculations. 

In  a  similar  manner,  the  connections  of  channel  purlins  to  the  web  of 
the  hip  or  valley  rafter  are  considered  in  Chapter  III,  followed  by  notes 
upon  other  styles  of  connection  in  Chapter  IV.  The  principal  formulas 
of  Chapters  II,  III  and  IV  are  derived  in  Chapter  V,  and  Chapter  VI 
shows  the  graphic  method  of  determining  the  principal  angles  of  any  of 
the  above  cases.  At  the  end  of  the  book,  tables  are  given  to  assist  in  the 
solution  of  the  problems  which  are  most  likely  to  occur  in  practice. 

By  means  of  this  arrangement,  a  man  may  obtain  all  the  desired  in- 
formation from  the  proper  general  drawing  and  the  corresponding  formulas 
without  the  necessity  of  reading  either  the  explanatory  notes  or  the  deriva- 
tions, although  it  is  recommended  that  he  read  them  at  least  once  when  he 
uses  the  system  for  the  first  time. 

Working  Lines.  —  A  few  working  lines  must  be  chosen  at  the  outset,  and 
all  details  must  be  referred  to  these  lines.  In  laying  out  work  it  is  often 
convenient  to  work  to  the  intersection  of  the  roof  planes,  or  to  the  inter- 
section of  the  planes  through  the  tops  of  the  mam  rafters.  It  is  well, 
however,  to  change  to  the  center  line  of  the  top  flange  of  the  hip  or  valley 
rafter  before  commencing  the  details.  This  may  be  readily  accomplished 
by  using  the  vertical  distance  v.  It  is  customary  to  work  either  to  the 
center  lines  or  to  the  faces  of  the  columns  or  main  rafters,  but  this  depends 
upon  the  conditions  of  the  particular  problem  and  does  not  affect  either  the 
principle  or  the  calculation,  after  certain  parts  are  chosen  at  the  beginning. 

In  general,  the  slope  of  each  roof  will  be  given,  and,  with  one  side  as- 
sumed (b',  b"  or  e),  the  remaining  values  may  be  determined.  But  fre- 
quently the  rafter  will  connect  to  columns  already  spaced,  which  means 
that  both  b'  and  b"  are  fixed,  and  that  the  slope  of  only  one  roof  is  given, 


in  which  case  the  slope  of  the  other  roof  must  be  calculated  to  correspond, 
by  transposing  formula  (2)  or  (5). 

Column  Connections.  —  The  connections  of  the  hip  or  valley  rafters  to 
the  columns  or  main  rafters  cannot  be  treated  fully  in  this  volume,  partly 
because  they  depend  upon  so  many  conditions  that  no  standard  detail  can 
be  given,  and  partly  because  any  draftsman  who  undertakes  to  detail  a  hip 
or  valley  rafter  should  experience  no  difficulty  in  designing  bent  plate  con- 
nections at  the  ends  of  the  rafter  since  they  generally  involve  only  the 
angles  which  he  has  already  determined.  However,  several  different  types 
are  represented  in  the  illustrative  problems  in  Chapter  II  and  it  is  hoped 
that  these  will  aid  the  novice  in  detailing  connections  for  his  particular 
rafter. 

Computation.  —  For  simplicity,  all  formulas  have  been  arranged  so  that 
they  involve  the  use  of  only  three  functions  of  any  angle,  namely  the  sine, 
the  cosine  and  the  tangent.  This  reduces  to  a  minimum  the  use  of  the  loga- 
rithmic tables  and  the  tabulation  of  logarithms,  as  well  as  the  possibility 
of  error  in  referring  to  either.  Furthermore,  the  average  draftsman  is 
doubtless  slightly  more  expeditious  in  the  use  of  these  three  functions  than 
in  the  use  of  the  others. 

It  is  assumed  that  a  book  of  tables  will  be  used  which  gives  the  loga- 
rithms of  dimensions  which  are  expressed  in  feet  and  inches  and  fractions  of 
inches.  Some  books*  give,  in  addition,  the  logarithms  of  all  the  common 
functions  of  angles  whose  tangents  are  expressed  by  slopes  in  inches  and 
fractions  per  foot.  The  use  of  such  tables  not  only  simplifies  the  loga- 
rithmic work,  but  often  precludes  the  necessity  of  finding  the  tangent  of 
an  angle  in  order  to  determine  the  slope. 

It  is  considered  superfluous  to  differentiate  between  logarithms  and 
cologarithms,  since  the  latter  may  easily  be  written  directly  from  the 
former  by  subtracting  each  figure  from  nine,  except  the  right-hand  one, 
which  is  taken  from  ten. 

Consistent  Accuracy.  —  Although  the  degree  of  precision  must  be  deter- 
mined to  satisfy  the  individual  requirements,  yet,  in  most  work  of  this 
nature,  all  linear  dimensions  should  be  expressed  to  the  nearest  sixteenth 
of  an  inch.  To  accomplish  this  purpose,  five  place  logarithms  should  be 
used,  those  for  the  functions  of  angles  being  interpolated  for  seconds. 

*  Smoley's  Tables  of  Logarithms  and  Squares,  McGraw-Hill  Book  Co.,  New  York, 
are  recommended. 


i  II  Al'TER  I.     CENERAL  OUTLINE 


3 


Notation. — All  values  which  deal  with  the  main  rafter  i>r  purlins  of  the 
steeper  roof  bear  tliis  mark  (').  Those  dealing  with  the  parts  of  the 
other  roof  bear  this  murk  (")•  Values  which  apply  to  parts  of  both 
roofs,  or  to  the  hip  or  valley  rafter,  are  represented  by  the  simple  letter 
without  a  diMinguishing  mark.  Angles  are  represented  by  capital  letters, 
and  their  slopes,  or  the  tangents  of  the  angles  in  inches  for  a  base  of  one 
foot,  by  the  corres]H>nding  lower  case  letters.  Distances  are  expressed  by 
lower  c:i<e  letters.  As  far  as  possible,  the  first  part  of  the  alphabet  has 
been  used  either  for  given  values  or  for  values  easily  found,  leaving  the 
last  part  of  the  alphaln't  for  the  more  difficult  and  important  parts  required. 


The  notation  employed  in  the  algebraic  solution,  Chapters  II,  III,  and 
l\,  is  given  below,  with  sufficient  explanation  for  identification,  and 
for  aid  in  the  interpretation  of  the  drawings.  Note  that  the  letters 
referring  to  the  flatter  roof  are  omitted,  on  account  of  the  similarity 
between  those  bearing  the  mark  (")  and  those  of  the  steeper  roof  bearing 
the  mark  ('). 

An  independent  system  of  letters  is  used  in  the  analyses  of  the  graphic 
method,  Chapter  VI,  but  all  anyles  bear  the  same  letters  as  the  correspond- 
ing angles  of  the  algebraic  method. 


NOTATION 

A'  =  the  angle  of  inclination  of  the  steeper  slope.  q' 

a'  =  the  slope  of  the  steeper  roof.  r' 

b'  =  the  hori/ontal  distance  between  the  working  points  of  the  steeper 

roof.  s' 

C  =  the  horizontal  angle  between  the  axis  of  the  steeper  roof  and  the 
hip  or  valley  rafter. 

C    =  the  tangent  of  the  angle  C. 

d'  =  the  slope  distance  between  the  working  points  of  the  steeper  roof. 

e    =  the  vertical  distance  between  the  working  points  of  the  rafters. 

f  =  the  width  of  the  top  flange  of  the  hip  or  valley  rafter. 

g'  =  the  gage  in  the  purlins  of  the  steeper  roof. 

H  =  the  angle  of  inclination  of  the  hip  or  valley  rafter. 

h    =  the  tangent  of  the  angle  //. 

{'   =  the  horizontal  distance  measured  along  the  purlin.  (See  page  41.) 

j'    =  the  horizontal  di>tance  measured  along  the  rafter.    (Seepage41.) 

it'    =  the  horizontal  distance  along  the  line  of  bend  corresponding  to  u'- 

L   =  the  angle  between  the  axes  of  the  two  roofs. 

m  =  the  horizontal  distance  between  the  working  points  of  the  hip  or 

valley  rafter. 

n   =  the  slope  distance  between  the  working  points  of  the  hip  or 
valley  rafter. 


=  the  horizontal  distance  along  the  purlin  of  tfie  steeper  roof,  from 
the  intersection  of  the  purlin  plane  and  the  center  line  of  the 
hip  or  valley  rafter  to  the  working  point  at  the  upper  end  of 
the  hip  or  valley  rafter. 


t    = 


t' 


u'  = 


u'  = 


w'  = 

A"  = 

x'  = 

y'  = 

z'  = 


the  distance  along  the  steeper  main  rafter  corresponding  to  v. 
the  distance  along  the  steeper  main  rafter  from  the  back  of  the 

purlin  to  the  upper  end  of  the  main  rafter, 
the  slope  distance  along  the  hip  or  valley  rafter  corresponding 

top'. 
(Chapter  III)  the  thickness  of  the  web  of  the  hip  or  valley 

rafter. 
(Chapter  IV)  the  thickness  of  the  plate  connecting  the  Tec  purlin 

of  the  steeper  roof  to  the  hip  or  valley  rafter. 
(Chapter  II)  the  distance  normal  to  the  steeper  roof  corresjxmd- 

ing  to  v . 
(Chapter  III)  the  normal  distance  from  the  top  of  the  steeper 

main  rafter  to  the  top  of  the  purlin, 
the  vertical  distance  which  the  top  of  the  hip  or  valley  rafter  is 

lowered  below  the  planes  through  the  tops  of  the  main  rafters. 

(See  page  19.) 
the  tangent  of  the  angle  between  the  line  of  bend  and  the  center 

line  of  the  top  flange  of  the  hip  or  valley  rafter.    (Steeper  roof.) 
the  angle  of  the  bend  in  the  connection  plate.     (Steeper  roof.) 
the  tangent  of  the  angle  A"', 
the  tangent  of  the  angle  between  the  line  of  bend  and  the  flange 

of  the  purlin  of  the  steeper  roof, 
the  tangent  of  the  angle  }>etwecn  the  purlin  of  the  steeper  roof 

and  the  hip  or  valley  rafter  in  the  plane  of  the  roof. 


CHAPTER  II 
FLANGE  CONNECTION 

FORMULAS  FOR  HIP  RAFTER  CONNECTIONS 
CASE  I  (a).     (Ordinary  Case.) 

Channel  purlins  connecting  to  flange  of  hip  rafter. 
Axes  of  roofs  intersecting  at  right  angle. 
Unequal  pitches. 

Given:  —  a'  =  the  slope  of  the  steeper  roof  (a'  always  greater  than  a"). 

a"  =  the  slope  of  the  other  roof. 

b'  =  the  horizontal  distance  between  the  working  points  of  the  steeper  roof.      (Either  b"  or  e  might  be  given  instead.) 

/  =  the  width  of  the  flange  of  the  hip  rafter. 
g'  and  g"=  the  purlin  gages, 
r'  and  r"  =  the  distances  from  the  working  points  to  the  backs  of  the  purlins,  measured  along  the  tops  of  the  main  rafters. 


(1) 

tan  A'  = 

a'. 

(2) 

e  = 

b'  tan  A'. 

(3) 

rl' 

b' 

cos  A' 

(4) 

tanA"  = 

a". 

(5) 

h"  — 

e 

v      ~~ 

tan  A"' 

(6) 

d" 

b" 

cos  A"' 

(7) 

tanC   = 

tan  A" 

tan  A'  ' 

(8) 

c    = 

tanC. 

(9) 

tan#   = 

tan  A'  sin  C. 

(10) 

h   = 

tan#. 

(11) 

b' 

sinC 

m 


\1*J 

11  —  —  j-.  • 
cos// 

*(13) 

v  =  ^tan4'cosC+ 

4 

(14) 

w'  =  v  cos  A'. 

(15) 

q'=  v  sin  A'. 

(16) 

,     (r'-q')cosA' 

tanC 

(17) 

„,_         P' 

cos  C  cos  H 

t(18) 

.     tan  C  cos  H 

"W  — 

cos2  A' 

1(19)    s 

in  X'  =  smA'  cos2  C  cos  H  . 

(20) 

x'=  tanZ'. 

-fa*  may  be  added,  if  necessary,  to  eliminate  sixteenths  from  the  value  of  v. 


(21) 
(22) 

(23) 
(24) 

(25) 
t(26) 


y'  =  w'sinX'. 

u"  —  v  cos  A". 

q"=  vsmA". 

p"  =  (r"-q")cosA"ia,iLC. 

S"=         P" 

sin  C  cos  H 

cos2  A  "tan  C 


cosH 

(27)  sin  X"  =  sin  A  "  sin2  C  cos  H. 

(28)  *"  = 

(29)  y"  = 


t  If  the  value  of  w'  or  w"  is  greater  than  I'D"  the  bevel  should  be  reversed  on  the  drawing  so  that  the  longer  side  becomes  the  12"  base  and  the  shorter  side 
w7  OT  to7'      These  values  are  obtained  directly  from  the  cologarithms.     Care  should  be  taken,  however,  that  the  original  values  of  w'  and  w"  are  used  in  formulas  (21)  and  (29). 

4 


rilAl'TKK  II.     FIANCE  CONNMTION 


<•< 


NOTES 

1  =StaTidar«i  rare  top  flange  of  hip  rafter 
n  1  JII— Make  •ufnrU-iiUj  lantr  U>  all. .w  rlr.-t«  or 

bolU  to  be  ptaeed  kfter  plate  U  bent . 
IT  A  V=All-.w  infflrleot  ectee  dUtanre  In  porlln 
when  cutiquare  at  center  of  hip  rafter. 
VI A  VH» Determine  dUunrr  e.  la  e".  bolm  by  larout 
Hake  aa  larce  M  practical. 
Wi.llh  of  plate  mar  be  ma 
than  that  of  flancr  If  deatred. 
Kwp  wkltb  In  even  Incbei  U  p 


CASE  I  (a)    HIP  RAFTER 
See  opposite  page. 


Fio.  2. 


6 


HIP  AND  VALLEY  RAFTERS 


FORMULAS  FOR  VALLEY  RAFTER  CONNECTIONS 

CASE  I  (6).     (Ordinary  Case.) 

Channel  purlins  connecting  to  flange  of  valley  rafter. 
Axes  of  roofs  intersecting  at  right  angle. 
Unequal  pitches. 

Given:  —  a'  =  the  slope  of  the  steeper  roof  (a'  always  greater  than  a"), 
a"  =  the  slope  of  the  other  roof. 

b'  =  the  horizontal  distance  between  the  working  points  of  the  steeper  roof.     (Either  b"  or  e  might  be  given  instead.) 
/  =  the  width  of  the  flange  of  the  valley  rafter. 
g'  and  g" '=  the  purlin  gages. 
/  and  r"  =  the  distances  from  the  working  points  to  the  backs  of  the  purlins,  measured  along  the  tops  of  the  main  rafters. 


(1) 

tan  A'  = 

a'. 

(2) 

e   = 

V  tan  A 

(3) 

d'  = 

b' 

cos  A' 

(4) 

tanA"  = 

a". 

(5) 

b"  = 

e 

tan  A" 

(6) 

d"  = 

b" 

cos  A" 

n^ 

n  r   - 

tan  A" 

.(8) 

(9) 

(10) 

(11) 


tan  A' 
c   —  tan  C. 

tan  H   =  tan  A'  sin  C. 
/i   =  tan  H. 
b' 


m   = 


sin  C 


(12) 

*(13) 

(14) 
(15) 

(16) 

(17) 

t(18) 

(19) 
(20) 


n  = 


m 
cosH 


f  1' 

v  =  *:  tan  A'  cos  C  +  T 

w'=  DCOS  A'. 
g'=  i)  sin  A'. 
,_  (/  —  q'}  cos  A' 
tanC 


s  = 


P' 


cos  C cos  H 

,     tan  C  cos  H 
w  =  -    — ^r, — 
cos2  A 

sin  X'  =  sin  A'  cos2  C  cos  H. 


(21) 
(22) 
(23) 
(24) 

(25) 

t(26) 

(27) 
(28) 

(29) 


y'  =  w'  sin  X'. 

u"=vcosA". 

q"=vsmA". 

p"=  (r"-3")cosA"tan<7. 

S"=         P" 

sin  C  cos  H 

cos2  A"  tan  C 

w"  =  -       — ^ 

cosH 

sin  X"  =  sin  A"  sin2  C  cos  H. 
x"=tanX". 


y"  = 


sinX" 
w" 


*  -fa"  may  be  added,  if  necessary,  to  eliminate  sixteenths  from  the  value  of  v. 
This  value  of ;;  gives  sufficient  clearance  to  permit  the  extension  of  the  purlin  beyond  the  valley  rafter.     Ordinarily  the  purlin  may  be  cut  at  the  center  line  of  valley  rafter, 

1" 
in  which  case  it  is  sufficient  to  assume  that  value  of  v  which  will  give  the  desired  clearance  at  the  center  line,  as,  for  example,  v  =  ^  .     Moreover,  if  the  purlin  is  cut  back  far 

enough  to  allow  ample  clearance  due  to  the  slope  of  the  valley  rafter,  then  v  becomes  zero,  and  formulas  (13)  to  (16)  and  (22)  to  (24)  are  reduced  to  the  following  forms: 


(13a)     v  =  0. 


(14a) 


(15a)    g'  = 


(16a)    p'  = 


r'cosA' 
tanC 


(22a)    u"  =  0. 


(23a)     q"  =  0. 


(24a)    p"  =  r"  cos  A"  tan  C. 


f  If  the  value  of  w'  or  w"  is  greater  than  1'  0"  the  bevel  should  be  reversed  on  the  drawing  so  that   the   longer  side  becomes  the  12"  base  and  the  shorter  side 

Care  should  be  taken,  however,  that  the  original  values  of  w'  and  w"  are  used  in  formulas  (21)  and  (29). 


— -,  or  — 77 .     These  values  are  obtained  directly  from  the  cologarithms. 


CHAP'ICl:    l[      I  I  AM1E  CONNECTION 


NOTES 

I  =  Standard  «••«  top  Bailee  of  Taller  rafter 
n  4  III  =  Make  raffle Irntlr  lante  to  allow  rtri-la  or 

bolu  to  be  placed  after  plate  l>  bent 
IV  *  V  —  Allow  •orlli-ient e<lK<-  cll«tancc  in  purlin. 

Sec  alao  followloc  note. 

VI  *  VII  =  Arrange  platea  with  edge*  normal  to  the 
line  of  bend.  Kerp  width  In  eren  Incbea 
Determine  wLltti  from  (paclnc  ol  holea 
forr.  n.  flace  hole*  tor 

pnrllnconncvUon  approximately 
I  .  -,  :.      -_ 


CAM  I  (6)    VAIJJSY  RAFTER 
See  oppoaitc  page. 


Fia.  3. 


HIP  AND  VALLEY  RAFTERS 

FORMULAS   FOR  HIP  RAFTER  CONNECTIONS 
CASE  II  (a).     (General  Case.) 

Channel  purlins  connecting  to  flange  of  hip  rafter. 
Axes  of  roofs  intersecting  at  oblique  angle. 
Unequal  pitches. 

Given:  —  a'  =  the  slope  of  the  steeper  roof  (a'  always  greater  than  a"), 
a"  =  the  slope  of  the  other  roof. 

b'  =  the  horizontal  distance  between  the  working  points  of  the  steeper  roof.     (Either  b"  or  e  might  be  given  instead.) 
/  =  the  width  of  the  flange  of  the  hip  rafter. 
g'  and  g"  =  the  purlin  gages. 

r'  and  r"  =  the  distances  from  the  working  points  to  the  backs  of  the  purlins,  measured  along  the  tops  of  the  main  rafters. 
L   =  the  angle  between  the  axes  of  the  two  roofs. 


(1) 

(2) 
(3) 
(4) 
(5) 

(6) 
(30) 

(31) 
*(8) 

tan  A'  = 
e  = 
ji 

a'. 
b'tanA'. 
b' 

(10) 
(11) 

(12) 

t(13) 

(14) 
(15) 

(16) 
(17) 

*njo 

h 
m 

n 

V 

u' 

q' 

p' 

s' 

in' 

=  tanH. 
b' 

(19) 
(20) 
(21) 
(22) 
(23) 

(32) 
(33) 

*(84) 

(35) 
(28) 

C2Q1 

sin  X'  = 

x'  = 

»'  = 
«"- 

3"  = 

•n" 

sin  A'  cos-  C  cos  H. 
tsmX'. 
w'smX'. 
vcosA". 
vsmA". 
(r"-q")cosA" 

sinC 

TO 

tanA"  = 

7," 

cos  A' 
a", 
e 

cosH 
f                          i" 

i  ton    A  '  nnc  C1     \ 

/7" 

tan  A" 
b" 

Z                                              4 

=  vcosA'. 
=  vsinA'. 
(r'  -q')  cos  A' 

P    ' 
*" 

tan  (L  -  C) 
P" 

fnn  (~l   — 

cos  A"' 
6'sinL      when  LOO" 

w"  = 

sinX"  = 
x"  = 

11"- 

cos  (L  —  C)  cos  H 
cos2  A" 

tanC  = 

c  = 

b"  +  b'  cos  L" 
b>  sin  L      when  L  >  00° 

tanC 
P' 

tan  (L  —  C)  cos  H 
smA"cos*(L-C)cosH. 
tanX". 
smX" 

—  b  cos  L 
tanC. 

cos  C  cos  H 
tan  C  cos  H 

*  If  any  of  these  values  exceeds  1'  0"  the  bevel  should  be  reversed  on  the  drawing  so  that  the  longer  side  becomes  the  12"  base  and  the  shorter  side  the  reciprocal 
of  the  value  found.     This  reciprocal  is  obtained  directly  from  the  cologarithm.     Care  should  be  taken,  however,  that  the  original  values  are  used  in  all  further  calculation, 
t  iV  may  be  added,  if  necessary,  to  eliminate  sixteenths  from  the  value  of  v. 


CHAPTKH   II.     FLANGK  CONNECTION 


.      NOTES; 
»  «««<Urd  r**t  top  tt*f,  ol  Up  raftor. 

»  MtVfaMljr  U*|»  to  aDow  rlnu  or 
koto  to  to  ftomi  aftor  yteto  b  toot. 

VI  *  TOIkMnlx  4Mu«  ^  u  ..  kob.  kj  kroM. 


WUU  of  pUir  m.,  to  • 
tktt  of  I«M>  »  <i~id. 


KEY  PLAN 


CASE  II  (a)     HIP  HA 
See  opposite  page. 


Fio.  4. 


10 


HIP  AND  VALLEY  RAFTERS 


FORMULAS  FOR  VALLEY  RAFTER  CONNECTIONS 
CASE  II  (6).     (General  Case.) 

Channel  purlins  connecting  to  flange  of  valley  rafter. 
Axes  of  roofs  intersecting  at  oblique  angle. 
Unequal  pitches. 

Given:  —  a'  =  the  slope  of  the  steeper  roof  (a'  always  greater  than  a"). 

a"  =  the  slope  of  the  other  roof. 

b'  =  the  horizontal  distance  between  the  working  points  of  the  steeper  roof.     (Either  b"  or  e  might  be  given  instead.) 

/  =  the  width  of  the  flange  of  the  valley  rafter. 
g'  and  g"  =  the  purlin  gages. 
r'  and  r"  =  the  distances  from  the  working  points  to  the  backs  of  the  purlins,  measured  along  the  tops  of  the  main  rafters. 

L   =  the  angle  between  the  axes  of  the  two  roofs. 


(1)  tan  A'  =  a'. 

(2)  e   =b'  tan  A'. 


cos  A'' 
(4)    tanA"=a". 

f 

' 


(10) 


fe=tantf. 
b' 


(19)  sin  X'  =  sin  A'  cos2  C  cos  H. 

(20)  x'  =  tanX'. 
y'=w'smX'. 


m 
cosfl' 


(15) 


(22) 
(23) 

(32) 
raa^ 

«"  = 
«"= 

<?"  — 

v  cos  A". 
vsinA". 
(r"  —  q")  cos 

A  " 

tan  (L  -  C 

') 

,Q1>.  ,,  6'sin.L 

(31)    tanC   =  ,-77  —  n  --  rwhenL  >  wr. 

b"  -  b'  cos  L 

*(8)  C   =  tan  C" 

(9)    tanH   =  tan  A'  sin  C. 


*(18) 


_  _ 
cos  C  cos  H 

tanCcosH  - 
cos2  A' 


(35)     sinX"=  sin  A"  cos2  (L  -  C)cosH. 

(28)  x"  =  tan  X". 

(29)  v" 


w 


*  If  any  of  these  values  exceeds  1'  0"  the  bevel  should  be  reversed  on  the  drawing  so  that  the  longer  side  becomes  the  12"  base  and  the  shorter  side  the  reciprocal  of 
the  value  ound.    This  reciprocal  is  obtained  directly  from  the  cologarithm.     Care  should  be  taken,  however,  that  the  original  values  are  used  in  all  further  calculation. 
t  -fg'  may  be  added,  if  necessary,  to  eliminate  sixteenths  from  the  value  of  v. 
This  value  of  v  gives  sufficient  clearance  to  permit  the  extension  of  the  purlin  beyond  the  valley  rafter.     Ordinarily  the  purlin  may  be  cut  at  the  center  line  of  valley 

rafter,  in  which  case  it  is  sufficient  to  assume  that  value  of  v  which  will  give  the  desired  clearance  at  the  center  line,  as,  for  example,  v  =  j   .     Moreover,  if  the  purlin  is  cut  back  far 
enough  to  allow  ample  clearance  due  to  the  slope  of  the  valley  rafter,  then  v  becomes  zero,  and  formulas  (13)  to  (16)  and  (22),  (23)  and  (32)  are  reduced  to  the  following  forms: 


(13a)     t>  =  0. 


(14a)    u'  =  0. 


(15a)    q'  =  0. 


(16a) 


(22a)    u"  =  0. 


(23a)     q"  =  0. 


(32a) 


-  C)  ' 


CHAPTI.l;    II.     FLANGE  CONNECTION 


11 


NOTES 

I-  Standard  nee  tap  fiance  of  rallcy  niter 
II  4  III-  Make  lufflclently  Urge  to  allow  rtreu  or 

boltt  to  be  placed  after  plate  U  bent 
IV  A  V-  Allow  •ufflclent  cxlce  dtetuce  la  purUn 

Sea  aim  following  note 

VI A  VII-  Arrance  ptatM  wlthedce*  normal  to  Uw 
Uncofbeod.  Keep  width  In  ercn  Incbe* 
Determine  width  from  (pacing  of  bole* 
JOT  rafter  connection.  Place  bole*  for 
'purlin  con neettop approi Imitely  oppoalU 


CASK  II  (6)    V ALLEY  RAITKR 
See  opposite  page. 


Fio.  5. 


12 


HIP  AND  VALLEY  RAFTERS 


FORMULAS  FOR  HIP  RAFTER  CONNECTIONS 
CASE  III  (a).     (Special  Case.) 

Channel  purlins  connecting  to  flange  of  hip  rafter. 
Axes  of  roofs  intersecting  at  right  angle. 
Equal  pitches. 

Given:  —  a   =  the  slope  of  each  roof. 

b   =  the  horizontal  distance  between  working  points,     (e  might  be  given  instead.) 

/  =  the  width  of  the  flange  of  the  hip  rafter. 
g'  and  g"  =  the  purlin  gages. 
r'  and  r"  =  the  distances  from  the  working  points  to  the  backs  of  the  purlins,  measured  along  the  tops  of  the  main  rafters. 

C   =  45°. 

c   =  12. 
9.84949   =  log  sin  45°  =  log  cos  45°. 


(36)      tan  A  =  a. 

e  =  b  tan  A . 
b 


(37) 
(38) 


cos  A 

(39)     tan  H  =  tan  A  sin  45°. 
h  =  tan  H. 
b 


(10) 
(40) 


(12) 


m  = 


n  = 


sin  45° 

m 
cosH 


*(41) 

(42) 
(43) 
(44) 
(45) 

(46) 


/  1" 

v   =     tan  A  cos  45°  + 


u   = 

q   =  v  sin  A  . 

p'  =  (r'  —  q)  cos  A. 

p"  =  (r"  -qjcosA. 


s   = 


(47) 
(48) 


w  = 


cos  45°  cos  H 
cos2  A 


cosH 

(49)  sin  X   =  sin  A  cos2  45°  cos  H. 

(50)  x   =  tanX. 


(51) 


cos  45°  cos  H 
'j*  may  be  added,  if  necessary,  to  eliminate  sixteenths  from  the  value  of  a. 


y  = 


w 


CAWS  III  (a)    Hir  RAFTER 
See  opposite  page. 


I V  *  V  >  Allow  lufflclent  edge  dliuncc  In  purlin 
»  boo  cut  Kjaan)  it  center  of  hip  rafter 
TI  *  >  II-  Dvtcnmne  dUUncK-.toc.  bole*  by  Ujom 
tUkru  Uicr  u  practical 
wi.ith  uf  pi«(e  majr  be  made  rreater 
Uuu  iiuitrf'i  ilaiiire  If  dnlrad 
Keep  wiUUi  In  I-TCU  Incnei  If  poolMe 


li..  .;. 


14 


HIP  AND  VALLEY  RAFTERS 


Given:  —  a   = 
b   = 

f  - 

g'  and  g"  •- 

r'  andr"  = 

C   -- 

c 
9.84949 

(36)  tan  A 

(37)  e 


FORMULAS  FOR  VALLEY  RAFTER  CONNECTIONS 
CASE  III  (b).     (Special  Case.) 

Channel  purlins  connecting  to  flange  of  valley  rafter. 
Axes  of  roofs  intersecting  at  right  angle. 
Equal  pitches. 

the  slope  of  each  roof. 

the  horizontal  distance  between  working  points,     (e  might  be  given  instead.) 

the  width  of  the  flange  of  the  valley  rafter. 

the  purlin  gages. 

the  distance  from  the  working  points  to  the  backs  of  the  purlins,  measured  along  the  tops  of  the  main  rafters. 
=  45°. 
=  12. 
=  log  sin  45°  =  log  cos  45°. 


(38) 


d  = 


(39)     tan  H 
(10)  h 


(40) 
(12) 


m  — 


n  = 


a. 

b  tan  A . 

b 
cos  A 

tan  A  sin  45°. 
tanH. 

b 
sin  45°' 

m 


cosH 


*(41) 

(42) 
(43) 
(44) 
(45) 

(46) 


/  1" 

v    =  -=  tan  A  cos  45°  +  T  • 

u  =  vcosA. 
q  =  v  sin  A . 
p'  =  (r'  —  q)  cos  A. 


(47) 
(48) 


P' 


w  = 


cos  45°  cos  H 
cos2  A 


P 


s'   = 


(r"  -  q)  cos  A. 

P' 
cos  45°  cos  H 


cosH 

(49)  sin  X   =sinA  cos2  45°  cos  H. 

(50)  x   =  tanX. 

smX 


(51) 


y  = 


w 


*  ^j"  may  be  added,  if  necessary,  to  eliminate  sixteenths  from  the  value  of  v. 
This  value  of  v  gives  sufficient  clearance  to  permit  the  extension  of  the  purlin  beyond  the  valley  rafter.     Ordinarily,  the  purlin  may  be  cut  at  the  center  line  of  valley 

1" 
rafter,  in  which  case  it  is  sufficient  to  assume  that  value  of  v  which  will  give  the  desired  clearance  at  the  center  line,  as,  for  example,  v  =  -r  •    Moreover,  if  the  purlin  is  cut  back 

far  enough  to  allow  ample  clearance  due  to  the  slope  of  the  valley  rafter,  then  v  becomes  zero,  and  formulas  (41)  to  (45)  are  reduced  to  the  following  forms: 


(41a)    v  =  0. 


(42a) 


0. 


(43a)     g  = 


(44a) 


=  r'cosA. 


(45a) 


r"cosA. 


(  HAITKK    II.      I  I, \.NG1     .  ONN1  <  T|M\ 


15 


T 
II 


NOTES 

=  Standard  mure  top  flance  of  rall.-r  ratter 
=  Make  »ufflrl,-nUr  larir,-  to  allow  rlreU  or 

bota  to  be  placed  afar  plateta  bent. 
TV  A  V= Allow  tnmclrnt  edffe  distance  In  purlin. 

Sec  alMi  (oUowlDK  note. 

Tl  4  VII  =  Arraoire  plate*  wim  eOtc*  normal  to  the 
line  of  bend.  Keep  width  In  eren  Inrnea. 
Determine  wldtn  frutn  apaclnc  uf  note* 
for  rafter  connection.  Place  bole*  for 
purllnconncrtlon 
tbennolc*. 


CASK  HI  (6)    VALLEY  RAITKB 

>.  ,    oppQ  .-.    ,.  ltr, 


Fio.  7. 


16 


HIP  AND  VALLEY  RAFTERS 


CALCULATION   FOR  HIP  AND  VALLEY  RAFTER  CONNECTIONS 

FLANGE  CONNECTION 

The  following  outline  is  given  to  serve  as  a  guide  for  the  tabulation  of  the  required  values,  and  to  indicate  the  logarithms  which  will  be  needed  for 
further  computation.     All  the  necessary  functions  of  an  angle  may  thus  be  determined  at  the  same  time. 

CASE  I.     (Ordinary  Case.) 

Roofs  at  right  angle;  unequal  pitches. 
Given:  —  a',  a",  V  (or  b"  or  e),  f,  g',  g",  r',  and  r". 


Num 
b'  - 

Angle. 

Slope. 

Logarithm. 

Logarithm. 

Sine. 

Cosine. 

Tangent. 

A' 

o'  = 

A" 

a"  = 

C 

c  = 

H 

h  = 

^x^ 

X' 

x'  = 

^x^ 

X" 

x"  = 

X 

ber.                   Logarithm. 

Number, 
v'  =  . 

e  =                            

s'  =  

d'  - 

w'  -    

&"=  

y'  =  • 

d"- 

u"=  

m  —  . 

q"=  

71     — 

v"=  . 

v  ~ 

s"  =  

u'  =  . 

w"-  . 

9'  = 


CHAITKK  II.     FLANGE  CONNECTION 


17 


CALCULATION   FOR  HIP  AND  VALLEY  RAFTER  CONNECTIONS 

FLANGE  CONNECTION 

The  following  outline  i~  mv.-n  t<>  >erve  as  a  guide  for  the  tabulation  of  the  required  values,  and  to  indicate  the  logarithms  which  will  be  needed  for 
further  computation.    All  the  necessary  functions  of  an  angle  may  thus  be  determined  at  the  same  time. 

CASE  II.     (General  Case.) 
Roofs  at  oblique  angle;  unequal  pitches. 
Oioen:  —  a',  a",  b'  (or  6"  or  e),  f,  g',  g",  rf,  r",  and  L. 


A' 


L- 


C- 


L-C- 


11 


X" 


o'- 


o"- 


c— 


ft- 


\n»ili,  r. 


V  = 

d'  = 
b"  = 
d"  = 
m  = 
n  = 

w'  = 

g'  = 


Logarithm. 


Ix^arithm. 


Number. 


,r       - 


P"  = 
«"- 

»"- 
»"- 


Logarithm. 


18  HIP  AND  VALLEY  RAFTERS 

CALCULATION   FOR  HIP  AND  VALLEY   RAFTER  CONNECTIONS 

FLANGE  CONNECTION 

The  following  outline  is  given  to  serve  as  a  guide  for  the  tabulation  of  the  required  values,  and  to  indicate  the  logarithms  which  will  be  needed  for 
further  computation.    All  the  necessary  functions  of  an  angle  may  thus  be  determined  at  the  same  time. 


CASE  III.     (Special  Case.) 
Roofs  at  right  angle;  equal  pitches. 


Given:  —  a,b  (or  e),  /,  g',  g",  r',  r",  C  =  45°. 


Nun 
b  - 

Angle. 

Slope. 

Logarithm. 

Sine. 

Cosine. 

Tangent. 

A 

a  — 

C 

c  =  12 

9.84949 

9.94949 

^x^ 

H 

h  = 

X 

X 

x  = 

^X^ 

iber.                   Logarithm. 

Number.                   Logarithm, 
n    =   ... 

G  — 

p'  = 

d  = 

p"-  

ffl    " 

s'  =  

n  — 

s"=  

V  '•  — 

ID     ~~ 

u  =  . 

11  =  . 

(  II A  IT  I  I!    11       FLANGE  CONNECTION 


19 


EXPLANATORY   NOTES  AND  SUGGESTIONS 


FLANGE  CONNECTION 


It  should  !«•  initial  that  in  every  case  the  hip  rafter  must  be  lowered  a 
certain  distance  r  to  allow  the  purlin  to  clear  the  flange  of  the  rafter. 
It  i~  not  necessary  to  lower  the  valley  rafter  provided  the  purlins  do  not 
extend  lieyi.inl  the  renter  line.  (See  note,  pp.  6,  10,  and  14.) 

Occasionally  it  may  l>e  preferable  to  prolong  the  purlin  past  the  center 
of  the  rnllry  rafter  if  a  better  connection  can  thus  \n-  obtained.  Sometimes 
purlins  are  supported  by  a  diagonal  rafter  when  neither  hip  nor  valley  is 
involved,  that  i-.  when  no  other  roof  intersects.  In  such  an  event,  the 
rafter  i-  lowered  a  distance  v  and  the  details  for  valley  rafters  used  in  prefer- 
ence to  tho-e  for  hip  rafters,  for  the  sake  of  appearance  and  strength. 

It  is  important  that  no  purlin  be  extended  past  the  renter  line  of  the  hip 
rafter,  for  if  prolonged  more  than  a  certain  distance  it  will  pierce  the  roof 
of  the  other  slope,  a  point  overlooked  by  some  authors  and  many  dr;r 
men.     This  distance  is  usually  very  small  and  is  hardly  worth  considering, 


connections  to  prevent  the  purlins  interfering  with  each  other,  and  to  pro- 
vide a  l>etter  connection  by  riveting  the  bent  plate  to  both  sides  of  the 
rafter  flange.  If  necessary,  however,  purlins  may  be  connected  at  the 
same  point  of  a  hip  rafter  by  means  of  independent  plates,  each  of  which 
is  fastened  by  two  rivets  through  one  side  of  the  rafter  flange,  provided 
the  bottom  flanges  and  the  bottom  part  of  the  webs  of  the  purlins  are  cut 
to  clear  each  other.  (See  Fig.  9,  p.  21.)  But  it  is  not  at  all  practical  to 
attempt  to  connect  two  purlins  to  a  valley  rafter  at  the  same  point  because 
of  extensive  interference. 

Connection  plates  are  kept  a  uniform  width,  partly  to  simplify  shearing, 
but  chiefly  to  avoid  the  reentrant  cut  which  is  expensive,  impractical,  and 
entirely  unnecessary,  although  many  dctailers  still  cling  to  it. 

It  is  suggested  that  plates  be  cut  parallel  to  the  center  lines  of  holes, 
not  only  for  the  sake  of  appearance,  but  also  to  enable  the  shopmen  to  cut 


i\    \    \    \    \    \ 


I •.,..  V 


in  view  of  the  fact  that  it  is  difficult  to  determine.  It  depends  upon  the 
pitches  of  the  roofs,  upon  the  depth  of  the  purlin  and  also  upon  the  width 
of  the  purlin  flange.  Ordinarily,  it  is  safer  to  cut  the  purlin  at  the  center 
line  of  the  hip  rafter  and  to  make  the  holes  in  the  connection  plate  corre- 
spond. This  accounts  for  the  apparently  distorted  form  of  the  plate. 

It  is  Ix'tter  not  to  connect  purlins  from  both  roofs  at  or  near  the  same 
point  of  the  hi])  or  valley  rafter.    They  should  have  entirely  separate 


a  complete  plate  with  one  stroke  of  the  shears  with  little  or  no  waste, 
except  at  the  end  plates.     (See  Fig.  8.) 

It  should  be  noticed  that  no  development  is  required  to  determine  the 
size  of  the  plate  for  the  connection  to  the  hip  rafter.  The  length  may  be 
found  by  adding  together  the  distances  from  the  line  of  bend  to  each  end 
of  the  plate,  measured  parallel  to  the  sides.  Yet  there  seems  to  be  less 
liability  to  mistake  in  the  shop  if  the  development  is  shown. 


20 


HIP  AND  VALLEY  RAFTERS 


ILLUSTRATIVE  PROBLEM 

CASE  I  (a).     HIP  RAFTEB 

Flange  connection;  buildings  at  right  angle;  unequal  pitches. 

Given  a  building  whose  roof  slopes  up  from  the  end  as  well  as  from  the  sides,  involving  two  hips  as  shown  in  the  Key  Plan,  Fig.  9.  One  end  of 
each  hip  rafter  will  connect  to  a  plate  and  angle  column,  and  the  other  to  the  peak  plate  of  one  of  the  main  trusses.  The  details  of  these  connections 
determine  the  working  points,  and  we  have :  — 

V  =  11'  1111"  =  12'  0"  -  Ty.  b"  =  14'  10|"  =  16'  0"  -  (6|"  +  7"). 

If  the  main  roof  is  j  pitch,  then  a"  =  6.     Using  a  12"  I  31 1#  rafter  and  8"u  11|#  purlins  we  have/  =  5,  and  g' '=  g"  =  3. 

The  required  values  are  given  below. 

(tor  necessary  computation  see  p.  24.) 


Number. 


9"=  H 

p'  =  3'  2V 
p'  =  8'  IIH" 


Logarithm. 
1.07862 


e  =  7'  5V 

0.87234 

d'  =  14'  If" 

b"=  14'  10!" 

1.17337 

d"=  16'  8" 

m  =  19'  H" 

1.28164 

n   =  20'  6TV 

• 

v  =  H 

9.09691 

0.50162 
0.95343 


Logarithm. 

Anglo. 

Slope. 

Sine. 

Cosine. 

Tangent. 

A' 

<*'  =  7A 

9.72272 

9.92900 

9.79372 

A" 

a"  =  6 

9.65051 

9.95154 

9.69897 

C 

c=9f 

9.79698 

9.89173 

9.90525 

H 

A=4H 

9.96930 

9.59070 

X' 

z'-3! 

9  47548 

9.49580 

X" 

z"-2 

9  21377 

9.21966 

Number. 
s'  =  4'  4TV 

s1  =  12'  4Ty 


«- 


i"=  H 


p"-7'2f" 

s"=  5'7|" 
s"=  12'4Ty 


Logarithm. 


w'  =  0.01655 


0.51401 
0.85860 


9.83903 


( HAITI:!!   II.      FIANGE  CONNECTION 


1'. 


S'u  ll<*' 

**'                .          »£ 

.   , 

,  '  . 

"i 

1  A 

**' 

,  . 

,,; 

<                            T't'' 

ft 

ILLCOTRATIVK  PROBLEM 
CAM  I  (a)    HIP  RAFTER 

>..    ..;;...-/.    j..,^. 


Fio.  9. 


22 


HIP  AND  VALLEY  RAFTERS 


ILLUSTRATIVE  PROBLEM 

CASE  I  (6).    VALLEY  RAFTER 

Flange  connection;  buildings  at  right  angle;  unequal  pitches. 

Two  intersecting  buildings  have  lean-to's  whose  roofs  form  a  valley,  as  shown  in  the  Key  Plan,  Fig.  10.  The  upper  end  of  the  valley  rafter 
connects  to  the  face  of  the  main  column,  and  the  lower  end  to  the  lean-to  column  outside  of  the  connection  of  the  main  lean-to  rafter.  The  pitches 
of  the  roofs  are  \  and  \  and  the  distance  from  the  center  line  of  the  lean-to  columns  supporting  the  steeper  roof  to  the  center  line  of  the  main 
columns  is  10'  0".  If  the  columns  are  85"  and  12j"  respectively,  back  to  back  of  angles,  we  have 

b'  =  9'  If"  =  10'  0"  -  (4|"  +  |"  +  6|"). 

Also  a'  =  6,  a"  =  4||,  and  g'  =  g"  =  3,  r'  =  r"  =  5'  0". 
The  required  values  are  given  below. 

(For  necessary  computation  see  p.  25.) 


Logarithm. 

Angle. 

Slope. 

Sine. 

Cosine. 

Tangent. 

A' 

a'  =  6 

9.65051 

9.95154 

9.69897 

A" 

a"=4H 

9.56983 

9.96777 

9.60206 

C 

c=9f 

9.79567 

9.89258 

9.90309 

H 

h  =  3l 

9.97979 

9.49464 

X' 

z'-3i 

9  41546 

9  43070 

X" 

z"-lti 

9  14096 

9  14515 

Number. 

Logarithm. 

V  =  9'1|" 

0.95974 

e   =4'6H" 

0.65871 

d'  =  10'  2TV' 

b"=  11'  4J" 

1.05665 

d"  =  12'  3  J" 

771     =   14'  7^" 

1.16407 

n   =  15'  3TV 

v   =  0 

u'  =  0     • 

Number. 

P'  =  5'  7Ty 
sr  =  r  e" 


u"=  0 


s"=  6'2f" 
w"=  8H 

y"= 


Logarithm. 
0.74742 

9.97980 


0.56983 
9.85884 


CHAPTER  II.     FLAN<  ,  I  <  T|.  >N 


23 


-    LEAN -TO 


PBOBLJUI 

I  (6)    VAI.LET  RAFTER 
Bee  opposite  page. 


.  10. 


24 


HIP  AND  VALLEY  RAFTERS 


ILLUSTRATIVE  PROBLEM 

CASE  I  (a). 
Computation  of  values  given 


6"  = 

tanA"  = 
e  •- 
b'  -- 
tan  A' 

b' 

cos  A' 
d' 


1 . 17337 
9.69897 
=  0.87234 
=  1.07862 
=  9.79372 

=  1.07862 
=  9.92900 
=  1 . 14962 


b' 
sin  C 

m 
cos  H 


1.07862 
9.79698 
1.28164 
9.96930 


n  =  1.31234 


f/2 

tan  A' 
cos  C 


b"-- 
cos  A"  = 

d"=  1.22183 


1.17337 
9.95154 


9.31876 
9.79372 
:  9. 89173 
=  9.00421 


tan  A"  =9. 69897 
tan  A'  =  9.79372 
tanC  =  9.90525 


v 
cos  A' 


9.09691 
9.92900 


u'  =  9.02591 


tan  A' 
sin  C 
tanH 


=  9.79372 
=  9.79698 
=  9.59070 


v 
sin  A' 


9.09691 
9.72272 


q'  =  8.81963 


:s  given 

r'  -q' 
cos  A' 
tanC 

P' 
cos  C 
cosH 
•'  s' 

tan  (7 
cosH 
cos2  A' 
w' 
\/w' 

sin  A' 
cos2C 
cos  H 
sinX' 
w' 

y' 

v 
cos  A" 

on  p 

=  0. 
=  9. 
=  0. 

.20. 

47787    0.92968 
92900    9  .  92900 
09475    0.09475 

v  = 
sinA"  = 

q"= 

r"  -  q"  = 
cosA"  = 

tan  C 

9 
9. 

.09691 
65051 

1 
9 
9 

00181 
.95154 
.90525 

8. 

0. 
9 

9 

74742 

65722  ' 
.95154 
.90525 

=  0. 
=  0. 
=  0. 

50162    0.95343 
10827    0.10827 
03070    0.03070 

64059    1.09240 

90525 
96930 
14200 

P"  = 

sin  C  = 
cosH  = 

s"  = 

cos2A"  = 
tanC  = 
'cosH  = 
w"  = 

oi  n  A" 

. 

=  9. 
=  9 

=  o' 

0 
0 
0 

.51401 
.20302 
03070 

0 
0 
0 

.85860 
.20302 
.03070 

0 

9 
9 

0 

.74773 

90308 
90525 
.03070 

1 

.09232 

=  0. 

=  9. 

=  9. 
=  9. 
=  9. 

01655 
98345 

72272 
78346 
96930 

9 

9 
9 
9 

83903 

,59396 
.96930 

=  9. 
=  0. 

47548 
01655 

sin2C  = 
cosH  = 

smX"  = 

w"  = 

y"  = 

49203 

09691 
95154 

. 

=  9 
=  9! 

9 
9 

,21377 
,83903 

9 

.37474 

Additional  calculation  required  for  the  above  problem  is  as  follows:  — 
At  peak,  to  determine  r'  and  r",  etc. 

T-V   =8.19382 
cos  A'  =  9.92900 


M"=9.04845 


7   =9.76592 
tan  A"  =9. 69897 


J  =8.26482 
/  =  3'  o|"  =  3' : 

Bent  plate  at  peak: 


-1" 


3|   =9.46489 


web  =  A  =  8*19382 

tan  C  =  9.90525 

1  =  8.28857 


//2  =  9  .  31876 
tan  C  =  0  .  09475 
cos  H  =  0  .  03070 


|  web  =  T3j? 
sin  C 


=  8.19382 
=  9.  79698 


iweb  =  T\ 
tanC 


=  8.19382 
=  9.90525 
=  8.09907 


tanC  =9.90525 

cos  A'  =  9.92900 

111  =9.97625 


7  =  9.76592 
cos  A"  =9. 95154 

7  it    =9.81438 

r"=4'7tV"  =  5'3"-71f". 
Bent  plate  at  column: 

iweb  =  -ft    =8.19382 
cos  C   =  9.89173 
J    =8.30209 
Cut  in  purlin  flange: 

tanC   =9.90525 
cos  A"  =9. 95154 


Cut  in  rafter  flange: 

//2  =  9. 31876 
tan  C  =  9. 90525 
cos  H  =  0.03070 
Say  3}".  2T3F  =  9.25471    Say  2£". 

Note.  —  No  distinction  is  made  between  logarithms  and  cologarithms,  since  this  is  apparent  from  the  formulas. 


f   =9.85679 


(HAITI  K  ii.     FLANGE  CONNECTION 


26 


O.O.V.'Tl 

'.i  r,-.i  v  .7 


6' 

tan  A' 

tan  A" 

b"=  1.05665 
cos  A"  =9. 96777 

d"  =  1.08888 

6'  =0" 

cos  A'  =  9.95154 
d'  =  1.00820 

tan  A"  =9. 60206 
tan  A'  =  9.09897 
tanC  =9.90309 


ILLUSTRATIVE  PROBLEM 

CASE  I  (6). 
Computation  of  values  given  on  p.  22. 


tan  A' 
sin  C 

tan  // 

6' 
sin  C 

in 


••  89697 
9.79567 
9.49464 

0.95974 
9.79567 
1.16407 
9.97979 
1.18428 


r1  =  0.69897 

t*f\a    Af    —    O    Q "»  I  "»  1 

\*\JO    4\         ~~     «7  .   J«  /  1  i  (~t 

tanC  =0.09691 
p'  =  0.74742 


tot  r 


0.10742 
0.02021 

ii 


Additional  calculatbn  rccniircd  for  the  above  prol)lcm  is  as  follows: 
Bent  plates  at  columns: 

iweb  =  A  =  8. 11  It,} 
tanC  =  9.90309 
W  =  8.21155 


tanC 

.  r 


9.90309 
9.97979 
0.09692 


sin  A' 

,-,„  c 

COS// 

sin  .V 
w' 


tanC 

P" 

sin  (' 

DM  // 


•9.65051 
•9.78516 
9.97979 
9.41546 
9.97980 
9.39526 

0.69897 
9.96777 
9.90309 

(i  .-,»;-. iv; 
0.20433 
0.02021 


f" -0.79437 


sin  C 


8.11KH 
9.79567 
8.31897 


co«M"- 9.93554 

tanC   -9.90;i09 

cosH   -0.02021 

w"-  9.85884 

sin  A" -9. 56983 

sin'C   -9.591:; I 

cos//   -  9.97979 

sin  X"- 9. 14096 

u>"=  9.85884 

I/"-  9.28212 


Note.  —  No  distinction  is  made  between  logarithms  and  eologarithnw  since  this  is  apparent  from  the  formulas. 


CHAPTER    III 

WEB   CONNECTION 

FORMULAS  FOR  HIP  RAFTER  CONNECTIONS 

CASE  IV  (a).     (Ordinary  Case.) 

Channel  purlins  connecting  to  web  of  hip  rafter. 
Axes  of  roofs  intersecting  at  right  angle. 
Unequal  pitches. 

Given:  —  a'  =  the  slope  of  the  steeper  roof  (a'  always  greater  than  a"). 
a"  =  the  slope  of  the  other  roof. 

b'  =  the  horizontal  distance  between  the  working  points  of  the  steeper  roof.     (Either  b"  or  e  might  be  given  instead.) 
t   =  the  thickness  of  the  web  of  the  hip  rafter. 

u'  and  u"  =  the  perpendicular  distances  from  the  tops  of  the  main  rafters  to  the  tops  of  the  purlins. 
r'  and  r"=  the  distances  from  the  working  points  to  the  backs  of  the  purlins  measured  along  the  tops  of  the  main  rafters. 

(1)  tan  A'  =  a'.  m  (58)  V  =  u'y'. 

\\-6)  n fj~  ^ 

(2)  e   =  b'tanA'.  *(59) 


(52)  i'  = 


t/2  ~cosA' 


-=*•  =*'  (60)  i»-</2 


(4)    tanA''=a".  (53)  f -A 

(V           b"-      e                                                                              A>  <61)  /'-stanC. 
~teET>'                                        (ifia)         p'  =  rcosA. 

v,                                                                            tanC  (24a)  p"=  r"cosA"  tanC. 

(6)  d"=~c^A7''                                            (17)             S'  = ^ (2V  S"  =        p" 

tanA/,                                                                       cos C cos H  sin  C  cos// 

(7)  tanC   =  t^A/'                                            (54)           w,  ^  cos2 A' tan//  (62)  w>"  =  cos2  A "  tan2  C  tan  H. 

(8)  c   =tanC.  (63)  sin  X"  =  cos  A"  sin  C. 

(9)  tan//   =  tan  A' sin  C.                                    (55)     cosX' =  cos  A'cosC.  (64)  x"=tanX". 
(10)            h   =tan//.                                            *(56)            x'  =  tanX'.  (65)  j/"=  sin  A"tanC. 

6'                                                 ,,_               ,      sin  A'  (66)  k"=u"y". 

m  =  iuTC '                                                              y  ==  taHC  '  (67)  z"  =  cos  A"  tan  C. 

*  If  any  of  these  values  exceeds  1'  0"  the  bevel  should  be  reversed  on  the  drawing  so  that  the  longer  side  becomes  the  12"  base  and  the  shorter  side  the  reciprocal  of  the 
value  found.     This  reciprocal  is  obtained  directly  from  the  cologarithm.     Care  should  be  taken,  however,  that  the  original  values  are  used  in  all  further  calculation. 

26 


<  HAITI:!!  in.    \VI:H  <  t .\\KTION 


27 


CAMS  IV  (a)    HIP 
See  opposite  Doge 


I A  n  -Space  rirato  u  far  apart  a*  pncnemL 
III  *  1 V  -  PUec  rlreta  far  c  nougb  from  (Uncc 
to  fin  rafflclcnt  drlTlnc  clecnnce. 

{T  AUow  •raple  •*• <Uitance  to  "aru°- 

PUcc  hol«  "  «"  •««<  «  P«ct«<»l. 
iMTtocmffleleotedodMuiealn 
Plata.  Determine  IPKM  b/  lajout, 
PtarloK  bole*  In  Una*  normal  to  axil 
of  hip  rafter.  Cot  plate  parallel  to 
UM  Une  of  holM.  (See  Flc.  8  Pace  i«) 
*'  *  XII  -Place  botM  far  rn»iuih  from  line  of 
be  ml  to  allow  rtrcU  or  bolt*  to  be 
Placed.  Determine  dlmenaloiu  br 
laroot. 
XIII  *  XTV  =  Determine  br  larout. 


Km   11 


28 


HIP  AND  VALLEY  RAFTERS 


FORMULAS  FOR  VALLEY  RAFTER  CONNECTIONS 
CASE  IV  (6).     (Ordinary  Case.) 

Channel  purlins  connecting  to  web  of  valley  rafter. 
Axes  of  roofs  intersecting  at  right  angle. 
Unequal  pitches. 

Given:  —  a'  =  the  slope  of  the  steeper  roof  (a'  always  greater  than  a"). 
a"  =  the  slope  of  the  other  roof. 

V  =  the  horizontal  distance  between  the  working  points  of  the  steeper  roof.     (Either  b"  or  e  might  be  given  instead.) 
t   =  the  thickness  of  the  web  of  the  valley  rafter. 

u'  and  u"  =  the  perpendicular  distances  from  the  tops  of  the  main  rafters  to  the  tops  of  the  purlins. 
r'  and  r"  =  the  distances  from  the  working  points  to  the  backs  of  the  purlins  measured  along  the  tops  of  the  main  rafters. 


(1) 

tan  A'  = 

a'. 

(2) 

e   = 

V  tan  A'. 

(3) 

d'  = 

b' 

cos  .A' 

(4) 

tanA"  = 

a". 

(5) 

h" 

e 

0    — 

tan  A" 

b" 

(6) 

d"  = 

cos  A" 

(7) 

+  a  rt   C1 

tan  A" 

tan  u    — 

tan  A' 

(8) 

c   = 

tanC. 

(9) 

tan  H   = 

tan  A'  sin  C. 

(10) 

h   = 

tan#. 

/•in 

vn     = 

b' 

m 


k«; 

(52) 
(53) 
(16a) 
(17) 
(?A\ 

n 
i' 
f 

P' 
s' 

of 

cos  H 

t/2  . 

sin  C 

t/2 

tanC 
r'  cos  A' 

tanC 
P' 

cos  (7  cos  // 
cos2  A'  tan  H 

tan2C 

(55).    cos  X'  =  cos  A'  cos  C. 
*(56)  x'=tanZ'. 

sin  A' 


sinC 


'(57) 


y'  = 


tanC 


(58) 

k'  = 

u'y'. 

*(59) 

z'  = 

tanC 

cos  A' 

(60) 

i"  = 

t/2 

cos  C 

(61) 

/'  = 

^tanC. 

(24a) 

p". 

r"  cos  A"  tan  C. 

(25) 

„ 

P" 

sin  C  cos  H 

(62) 

u>"  = 

cos2  A"  tan2  C  tan  H. 

(63) 

smX"  = 

cos  A"  sin  C. 

(64) 

x"  = 

tanX". 

(65) 

?/'  = 

sin  A"  tan  (7. 

(66) 

k"  = 

«"*"• 

(67) 

z"  = 

cos  A"  tan  C. 

*  If  any  of  these  values  exceeds  1'  0"  the  bevel  should  be  reversed  on  the  drawing  so  that  the  longer  side  becomes  the  12"  base  and  the  shorter  side  the  reciprocal  of  the 
value  found.     This  reciprocal  is  obtained  directly  from  the  cologarithm.     Care  should  be  taken,  however,  that  the  original  values  are  used  in  all  further  calculation. 


'!!  \ITl.i:   111.     \\  I •:»  CONNECTION 


29 


•  Space  rtreU  w  far  apart  ai  practical. 

*  Place  rlToU  far  enoafheromflance  to 
(Ire  lufltcJcat  drlrlnc  dnrmnce. 

V  4  VI  =  Allow  ample  edfte  dldance  In  purlin. 

KlroU  mlKht  be  placed  In  a  line  paral- 
N  1 1   vim    '"' to  llne  °*  bcod  "  Preferred. 

•llowln*  (ancient  edge  dtotence  la 
Plato.  Detwmlno  <pacc(  bj  laroat, 
placliw  bolp«  In  a  llne  normal  to  axl* 
of  rallor  rafter.  It  It  preferable  to 
cat  plata  parallel  to  the  line  of  hnlM 


XI A  XII  -Place  bolea  tor  enooch  from  lino  of 
bond  to  allow  rtreu  or  bolu  to  to 
placed.  Determine  dlmeiuioiu  bj 
lajruut. 
XIII  &  XTY-I>elcnulno  bjr  Uyont. 


CAM  IV  (6)    VALLEY  RAPTOR 
See  opposite  page. 


Fio.  12. 


30 


HIP  AND  VALLEY  RAFTERS 


FORMULAS  FOR  HIP  RAFTER  CONNECTIONS 
CASE  V  (a).     (General  Case.) 

Channel  purlins  connecting  to  web  of  hip  rafter. 
Axes  of  roofs  intersecting  at  oblique  angle. 
Unequal  pitches. 

Given:  —  a'  =  the  slope  of  the  steeper  roof  (a'  always  greater  than  a"), 
a"  =  the  slope  of  the  other  roof. 

6'  =  the  horizontal  distance  between  the  working  points  of  the  steeper  roof.     (Either  b"  or  e  might  be  given  instead.) 
t   =  the  thickness  of  the  web  of  the  hip  rafter. 

u'  and  u"  =  the  perpendicular  distances  from  the  tops  of  the  main  rafters  to  the  tops  of  the  purlins. 

r'  and  r"  =  the  distances  from  the  working  points  to  the  backs  of  the  purlins  measured  along  the  tops  of  the  main  rafters. 
L   =  the  angle  between  the  axes  of  the  two  roofs. 

(1)  tan  A'  =  a'. 

(2)  e   =  b'  tan  A'. 

(3)  d'  =  c^A7' 

(4)  tan  A"  =  a". 

e    n 


(5) 
(6) 


b" 
d"  = 


tan  A" 
b" 


cos  A" 
(30)     tanC   =     , 


(31)     tanC   =  v, 

*(8)  c   =  tan  C. 

(9)    tantf   =  tan  A'  sin  C. 
(10)  h   =  tan  H. 

b' 


<  90°. 
when  L  >  90°. 


(ID 


m    — 


sinC 


(12) 
(52) 
(53) 
(16a) 

(17) 

*(64) 

(55) 
*(56) 

*(57) 
(58) 
*(59) 

m 

n  — 
„•/ 

cosH 

t/2 

•i 

sinC 

t/2 

3  — 
pf- 

Q' 

tanC 
r'  cos  A' 

tanC 
P' 

w'  = 

cosX'  = 
x'  = 
11' 

cos  C  cos  H 
cos2  A'  tan  H 

tan2C 
cos  A'  cosC. 
tan  X'. 
sin  A' 

y  - 

k'  = 

z'  = 

tanC 

u'y'. 
tanC 

cos  A' 

(68) 
(69) 

(32a) 
<yy\ 

i»-        </2 

sin  (L  -  C) 
r            t/2 

3       tan  (L  -  C) 
„       r"cosA" 

tan  (L  —  C) 
P" 

=  cos2  A"  tang 
*  tan2  (L  -  C)  ' 

(71)     sin  X"=  cos  A"  cos  (L  -  C). 

*ff\4.^  T" —  ton  V/; 

lU^r  I  JU      —"    I  i  i  1 1  ^\.      • 

sin  A" 


*(72) 

(66) 

*(73) 


tan(L-C)" 

cos  A" 
tan(L-C)' 


z   = 


*  If  any  of  these  values  exceeds  1'  0"  the  bevel  should  be  reversed  on  the  drawing  so  that  the  longer  side  becomes  the  12"  base  and  the  shorter  side  the  reciprocal  of  the 
value  found.    This  reciprocal  is  obtained  directly  from  the  cologarithm.    Care  should  be  taken,  however,  that  the  original  values  are  used  in  all  further  calculation. 


CHAPTER  III      WEB  CONNECTION 


31 


I  4  It  -  Spice  rlrcU  H  far  apart  u  practical. 
Ill  4  IV  =  Place  rlrcU  far  enough  from  flange 

to  giro  •udli-lent  driving  clearance. 
V  4  VI  =  Allow  ample  •dee  distance  In  purlin. 

KlToU  niUfht  be  placed  ID  •  line  paral- 
vn  *  vinl  lcl  *"  UDC  °'  bend  "  preferred. 
X  4  X     f  =  PUco  ***»  «  '«  »>«<  "  Pra 
'  allowlnif  .ufBclcnt  odge  dtatence 
In  plate.  Determine  IPBCM  by  lajout, 
placlnf  hole*  In  a  line  normal  to  axU  of 
hip  rafter.  It  U  preferable  to  cut  plate 
parallel  to  tbe  Unc  of  bulea.   (SeeFlc.8 
PaeelB.) 

II  *  XII  =  Place  bole*  far  enoueh  from  Urn  of 
bend  to  allow  rireU  or  bulu  to  be 
placed.  Determine  dlmenaloo*  bjr  layout. 
Till  A  XTV  =  Determine  b/  lajout. 


.        » 

u-3 


KEY  PLAN 


CAM  V  (a)    HIP  rUrncR 

. ppMlti  IMfft 


FJG.  13. 


32 


HIP  AND  VALLEY  RAFTERS 


FORMULAS  FOR  VALLEY   RAFTER  CONNECTIONS 

CASE  V  (6).     (General  Case.) 

Channel  purlins  connecting  to  web  of  valley  rafter. 
Axes  of  roofs  intersecting  at  oblique  angle. 
Unequal  pitches. 

Given:  — a'  =  the  slope  of  the  steeper  roof  (a'  always  greater  than  a"). 
a"  =  the  slope  of  the  other  roof. 

V  =  the  horizontal  distance  between  the  working  points  of  the  steeper  roof.     (Either  b"  or  e  might  be  given  instead.) 
I   =  the  thickness  of  the  web  of  the  valley  rafter. 

u'  and  «"=  the  perpendicular  distances  from  the  tops  of  the  main  rafters  to  the  tops  of  the  purlins. 

r'  and  r"  =  the  distances  from  the  working  points  to  the  backs  of  the  purlins  measured  along  the  tops  of  the  main  rafters. 
L   —  the  angle  between  the  axes  of  the  two  roofs. 


(1)  tan  A'  =  a'. 

(2)  e   =b'  tan  A'. 

d>  =coiA"'' 
(4)     tan  A"  =  a". 

o 

b"  = 


(5) 

(6)  d"  = 

(30)  tanC   = 

(31)  tanC   = 

*(8)  c  =tanC. 

(9)  tan#   =  tan  A' sin  C. 

(10)  h  =tantf. 
b' 


tan  A" 

V 
cos  A"' 

b'  sin  L 

b"  +b'cosL 

b'  sin  L 


when  L  <  90°. 
when  L  >  90°. 


(H) 


m   = 


sinC 


(12) 
(52) 
(53) 
(16a) 
(17) 

*(54) 

(55) 
*(56) 

*(57) 
(58) 
*(59) 

m 

;/ 

cosH 

t/2 

,v 

sinC 

t/2 

J  — 
_/ 

tanC 
r'cos  A' 

p 

(,' 

tanC 
t 

111' 

cos  C  cos  H 
cos2  A'  tan  H 

cosX'  = 
x'  = 

a- 

k'  = 

z'  = 

tan2C 
cos  A'  cos  C. 
tan  X'. 
sin  A' 

tanC 

u'y'. 
tan  C 

cos  A' 

(68) 
(69) 
(32a) 
(33) 

(70) 

(71) 
*(64) 

*(72) 

(66) 

*(73) 


t/2 


sin  (L  -  C) 
,,  </2 

J       tan(L-C)' 

„       /rcosA" 

P    "tan(L-C)' 

s"=  -  P" 

cos  (L  —  C)  cos  H 

cos2  A"  tan  H 
tan2  (L  -_C)  ' 

sinX"=cosA"cos(L-C). 
x"=tan,Y". 
sin  A" 


w"  = 


y 


tan  (L  -  C) 


k"=u"y". 


cos  A" 

tan(L-C)' 


*  If  any  of  these  values  exceeds  1'  0"  the  bevel  should  be  reversed  on  the  drawing  so  that  the  longer  side  becomes  the  12"  base  and  the  shorter  side  the  reciprocal  of  the  value 
found.    This  reciprocal  is  obtained  directly  from  the  cologarithm.     Care  should  be  taken,  however,  that  the  original  values  are  used  in  all  further  calculation. 


«  HAITI. i:   III.     \\KII  CONNECTION 


MOTES; 

fi»  ••  Car  •! 
TU  A  IV  -  PU«  rintobr  MOM*  h«i  «M«. 

V  4  n  •  Allow  u^to  •*««  dWur.  u  pgrite . 
BhralMlcto  W  phcW  to  •  111.  pud 

VII  \  1 1 1 '      ' 


IlUpralbrakbto 


V  (6)    VALLET  RAFTXB 
Bee  opporitc  pace. 


M  *  III  - 

xin  «  xiv  - 


«  !*»<•») 
koh.  (kr  «Mgk  (TO-  lb>  ot 
toalkiw  riM.  M  MM  to  b. 


I  ...    11. 


34  HIP  AND  VALLEY  RAFTERS 

FORMULAS  FOR  HIP  RAFTER  CONNECTIONS 
CASE  VI  (a).     (Special  Case.) 

Channel  purlins  connecting  to  web  of  hip  rafter. 
Axes  of  roofs  intersecting  at  right  angle. 
Equal  pitches. 

Given:  —  a  =  the  slope  of  each  roof. 

b   =  the  horizontal  distance  between  working  points,     (e  might  be  given  instead.) 
t   =  the  thickness  of  the  web  of  the  hip  rafter. 

u'  and  u"  =  the  perpendicular  distances  from  the  tops  of  the  main  rafters  to  the  tops  of  the  purlins. 

r'  and  r"  —  the  distances  from  the  working  points  to  the  backs  of  the  purlins  measured  along  the  tops  of  the  main  rafters. 
C   =  45°. 
c   =  12. 
9.84949   =  log  sin  45°  =  log  cos  45°. 

(36)  tan  A  =  a.  ^  {   ^     111  (78)  w  =  cos2Atanff. 

(37)  e  =  btanA.  sin45° 

f  (79)     sin  X   =  cos  A  sin  45°. 

(38)  d  =  — •  (75>  3   -I' 

cos  A  (80)  z=tanX. 

(39)  tan  H  =  tan  A  sin  45°.  (76)  Pr  =  r>  cos  A- 

(81)  y   =  sin  A. 

(10)  fe  =  tantf.  (77)  p"=r"cosA. 

5  ,  (82)  fc'-u'y. 

(40)  m  =  -^W  (46)  s'  =  -  -ng 5- 

sin  45°  cos  45°  cos//  (83)  k»=u»y. 

(12)  n--^.  (47)  ' 


cos  45°  cos  H  (84)  z   =  cos  A. 


rilAITKU   III.     WEB  CON  M.I  TH  >\ 


35 


NOTES 

I  •  Space  rirct»  u  tar  apart  u  practical. 
Ill -Place  rlrrU  far  rnouKh  from  Haute 
Co  tire  iufoclent  drlrlnir  clearance. 
V  *  VI  -  Allow  ami.ii-  cdire  dUtance  In  purlla 
RlrcU  micnt  be  placed  In  a  line  pnral- 
Icl  to  Uoe  of  bend  If  preferred. 
\  II.IX).Flacc  bokiai  far  apart  ai  practical, 
*  x)   allowing  •uftlck-nted(redl»Unce  In 
lOate.  Determine cpacei  by  layout, 
placing  bole*  In  a  line  normal  to 
ula  of  hip  rafter.  It  If  preferable  to 
cat  plate  parallel  to  the  lino  of 
bolra/Scc  Ft*.  8.  Pace  It) 
II  4  XII  •  Placcbulc*  far  enough  from  line 
of  bend  to  allow  rlreU  or  bolu  to 
be  placed.  Determine  dlmcnaloo*  br  layout 
XIII  •  Determine  br  layout. 


CAM  VI  (a)    HIP  RAFTER 


Fin    IS 


36  HIP  AND  VALLEY  RAFTERS 

FORMULAS  FOR  VALLEY   RAFTER  CONNECTIONS 
CASE  VI  (6).     (Special  Case.) 

Channel  purlins  connecting  to  web  of  valley  rafter. 
Axes  of  roofs  intersecting  at  right  angle. 
Equal  pitches. 

Given:  —  a   =  the  slope  of  each  roof. 

b   =  the  horizontal  distance  between  working  points,     (e  might  be  given  instead.) 
t   =  the  thickness  of  the  web  of  the  valley  rafter. 

u'  and  u"  =  the  perpendicular  distances  from  the  tops  of  the  main  rafters  to  the  tops  of  the  purlins. 
r'  and  r"  =  the  distances  from  the  working  points  to  the  backs  of  the  purlins  measured  along  the  tops  of  the  main  rafters. 

C   =  45°. 
c   =  12. 
9.84949   =  log  sin  45°  =  log  cos  45°. 

(36)      tan  A  =  a.  ,_,.  .          t/2  (78)  w=cos2AtanH. 

v       ~-    ~        j£O* 


(37)  e  =  btanA.  ^  (79)     ginX   =  cos  A  sin  45° 

'"*'    '  (80)  .-teZ. 


(39)  tan  H  =  tan  A  sin  45°.  (76)  p'=/cosA. 

^Oly  y      —    bill  -il. 

(10)  h  =  tan  tf .  (77)  P"  = 

(40)  «  =  ;io-  (46) 


(82)  k'  =  tt'y. 

0 


(g3)  fc/,=  wV 

(47)  S"=cos45PocosH'  (84)  2   =cosA. 


37 


NOTES 

I  -  Space  rlreU  u  tar  apart  w  practical  . 
m«  Place  rlreUlmr  fiiouch  from  nature 
to  a-lTeraOclentdrlrliur  clearance. 
V  A  VI  •  Allow  ample  edfo  dMancc  lu  purlin. 
BJrete  might  be  place*  In  •  line  pant- 
VII  ITI   lc"°  Uw>o">e<KH«  prefernxJ. 

*  i  f"  1>Uct>  hol<"  "  *"  "P"1  ""  P«c««<»l. 

•*|  »Mi.wli,g.amclenled«o<ll»Unccln 
pUte.  MHBIMBHH  br  Uroat, 
j>tacln>bokM  In  •  line  normal  to  uM 
of  Taller  rafter.  It  U  preferable  to 
cut  pUte  parallel  to  the  line  of 


CA«E  VI  (6)    VALUCY  IUrr»» 
See  oppmitc  page. 


„  .     _  .  , 

XI  *  XII-  Place  bolM  far  enoocn  from  line  o» 
bond  to  allow  rlroti  or  bolu  to  be 

Till-  KSrt""1"11  *'  'WU» 


Via.  10. 


38 


HIP  AND  VALLEY  RAFTERS 


CALCULATION  FOR  HIP  AND  VALLEY  RAFTER  CONNECTIONS 

WEB  CONNECTION 

The  following  outline  is  given  to  serve  as  a  guide  for  the  tabulation  of  the  required  values,  and  to  indicate  the  logarithms  which  will  be 
needed  for  further  computation.     All  the  necessary  functions  of  an  angle  may  thus  be  determined  at  the  same  time. 

CASE  IV.     (Ordinary  Case.) 
Roofs  at  right  angle;  unequal  pitches. 
Given:  —  a',  a",  V  (or  b"  or  e),  t,  u',  u",  r',  and  r". 


Nun 
V  - 

Angle. 

Slope 

Logarithm. 

Sine. 

Cosine. 

Tangent. 

A' 

a'  = 

A" 

a"  = 

c 

c- 

H 

h  = 

^x^ 

X' 

x'  = 

X 

X" 

x"  = 

X 

iber.                   Logarithm. 

Number.                  Logarithm, 
w'  =  

i>      — 

y'  -  • 

d'  - 

k'  =    

z'  =  

d"- 

j"-    

p"-    

i'  — 

s"=  

V  - 

, 

y"-  • 

s'  — 

k"-  

z"=  . 

CHAPTER  III.     WEB  CONNECTION 


CALCULATION   FOR   HIP  AND  VALLEY   RAFTER  CONNECTIONS 

WEB  CONNECTION 

The  following  outline  is  given  to  serve  as  a  guide  for  the  tabulation  of  the  required  values,  and  to  indicate  the  logarithms  which  will  be 
in •< •(!(•<!  for  furt  HIT  computation.    All  the  necessary  functions  of  an  angle  may  thus  be  determined  at  the  same  time. 

CASE  V.     (General  Case.) 
Roofs  at  oblique  angle;  unequal  pitches. 
»:  —  a',  a",  b'  (or  6"  or  «),  t,  u',  u",  r',  r",  and  L. 


A' 


A" 


L-C 


H 


X" 


Number. 


e  = 
d'  = 
6"  = 


m  = 
n  = 


o'- 


A- 


x'- 


Logarithm. 


Locarithm. 


Number. 
w'  -  ...... 


k'  - 
t'  - 


j"- 

„" 


Logarithm. 


v"- 


HIP  AND  VALLEY  RAFTERS 


CALCULATION   FOR  HIP  AND   VALLEY   RAFTER  CONNECTIONS 

WEB  CONNECTION 

The  following  outline  is  given  to  serve  as  a  guide  for  the  tabulation  of  the  required  values,  and  to  indicate  the  logarithms  which  will  be 
needed  for  further  computation.     All  the  necessary  functions  of  an  angle  may  thus  be  determined  at  the  same  time. 


CASE  VI.     (Special  Case.) 
Roofs  at  right  angle;  equal  pitches. 


Given:  —  a,  b  (or  e),  t,  u',  u",  r',  r",  and  C  =  45°. 


Nun 
b  = 

Angle. 

: 

Slope. 

Logarithm. 

Sine. 

Cosine. 

Tangent. 

A 

a  = 

C 

c  =  12 

9.84949 

9.84949 

^x^ 

H 

h  = 

^x^ 

X 

x  = 

x^ 

iber.                   Logarithm. 

Number.                   Logarithm, 
-n"-   

e  =  .    . 

s'  =  

d  - 

s"  -  

m.  = 

w   =  

?i  — 

v   —  . 

i  = 

k'  =  

7  =    . 

k"-  

7/=   . 

2    =    

UlAPTKK  III.     \Vi:il  CONN  K<  TION 


41 


EXPLANATORY  NOTES  AND  SUGGESTIONS 
WEB  CONNECTION 


e  the  purlins  connect  to  the  face  of  the  web  of  the  hip  or  valley 

rafter,  tin-  lines  <>f  liciul  of  the  connecting  plates  cannot  lie  in  the  same 

plain1  as  the  working  line  of  the  rafter,  which  is  the  center  line  of  the  top 

flange.     It  would  IM>  impractical  to  use  two  working  lines  in  the  planes  of 

the  well  face-,  and  so  the  line  of  bend  must  be  located  with  reference  to 

that  point  in  the  center  line  of  the  top  flange  which  is  cut  by  the  plane  of 

the  purlin  web.     The  horizontal  distance  from  this  working  point  to  the 

ction  of  the  line  of  bend  and  the  working  line  of  the  purlin  is  rep- 

<1  by  i'  or  i"  when  measured  along  the  purlin,  and  by  j'  or  j"  when 

red  in  the  direction  of  the  rafter.     The  distinction  between  i'  and 

;"  is  !.e-t  shown  in  Fig.  1 1 .  p.  27,  or  Fig.  12,  p.  29. 

It  is  generally  an  advantage  to  the  shop  to  have  the  rivets  and  holes  in 

iimection  plates  placed  in  lines  normal  to  the  flanges  of  the  beams  and 

channels,  and  this  can  usually  be  accomplished.     If  this  arrangement 

brings  the  holes  too  far  from  the  line  of  bend,  they  may  be  located  in  a 

line  parallel  to  the  line  of  lx>nd  instead. 

The  flange  of  the  purlin  may  be  sheared  diagonally  if  preferred,  but  in 


the  more  modern  practice  the  flange  is  blocked  out  as  shown.  The  slope 
(z*  or  z")  of  the  clearance  line  is  given  to  serve  l>oth  methods.  The  amount 
to  be  blocked  out  is  best  determined  by  a  layout,  with  allowance  for  any 
desired  clearance. 

The  connection  plate  should  l>e  placed  upon  the  obtuse  angle  side  of  the 
purlin  where  possible,  in  order  to  avoid  the  sharp  bend  in  the  plate  and 
also  to  facilitate  erection.  To  accomplish  this  in  the  case  of  the  valley 
rafter,  the  plate  must  be  put  on  the  upper  aide  of  the  purlin  web.  To 
retain  the  similarity  of  details  for  hip  and  valley  rafters,  the  purlins  are 
shown  with  their  flanges  facing  down  the  slope  instead  of  up.  Should  it 
be  desired  to  face  them  the  other  way,  see  Chap.  IV,  p.  48. 

In  order  to  simplify  erection  it  is  better  not  to  have  purlins  of  both 
slopes  connect  to  the  web  of  the  hip  or  valley  rafter  at  the  same  point. 
It  is  frequently  advisable  to  frame  them  opposite,  however,  and  this  may 
be  easily  accomplished  in  either  hip  or  valley  work,  by  arranging  the  holes 
in  I  Kith  connecting  plates  to  correspond  to  common  holes  in  the  rafter 
web. 


42 


HIP  AND  VALLEY  RAFTERS 


ILLUSTRATIVE  PROBLEM 
CASE  IV  (a).    HIP  RAFTER 
Web  connection;  buildings  at  right  angle;  unequal  pitches. 

For  comparison,  let  us  consider  a  hip  rafter  under  the  same  conditions  as  the  one  shown  on  pp.  20  and  21,  except  that  the  purlins  connect 
to  the  web  of  the  rafter  instead  of  to  the  flange.  The  working  lines  and  the  column  connections  remain  the  same,  and  it  is  unnecessary  to  reproduce 
the  latter  in  Fig.  17.  Given  V  =  11'  lltf",  6"  =  14'  101",  and  a"  =  6.  Using  a  12"  I31|#  rafter  and  8"ulli#  purlins,  we  have  t  =  |, 
u'  =  u"  =  2,r'  =  3'  Of"  and  r"  =  10'  1^  ".  The  required  values  are  given  below. 

(For  necessary  computation  see  p.  46.) 


Number. 
V  =  11'  lilt" 

e  =  r  5Ty 

d'  =  14'  If" 
fe"=  14' 10|" 
d"=  16' 8" 
m   =  19'  1|" 
n   =  20'  6Ty 

i'  =  & 
j>  =  i 

p'  =  3'  2T|" 
s'  =  4'  54" 


Logarithm. 

Sine. 

Cosine. 

Tangent. 

A' 

o'  =  7& 

9.72272 

9.92900 

9.79372 

A" 

a"  =  6 

9.65051 

9.95154 

9.69897 

C 

c=9| 

9.79698 

9.89173 

9.90525 

H 

A  =  4H 

9  96930 

9  59070 

X' 

^7=10  A 

9.82073 

0  05414 

X" 

z"=8| 

9.74852 

9  83039 

Logarithm. 
1.07862 
0.87234 

1.17337 
1.28164 


0.51130 


Number. 
w'  =  5T<V 

v'  =  71 
k>  =  IT% 

z'  =  ill 


r-  i 

p"=  7'  31" 


y"= 
fc"=  I 

2"=  81 


Logarithm. 
9.81747 


0.86107 


9.55576 


•  HAITI. i:  in      \\i.r,  <  ONN1  <  i  i"\ 


C  L    f!-.-..  r 


iLLDsnuTtvm  PROBLBM 

CAM  IV  (a)    HIP  RAITM 

See  opposite  page. 


Fio.  17. 


44 


HIP  AND  VALLEY  RAFTERS 


ILLUSTRATIVE  PROBLEM 

CASE  IV  (6).    VALLEY  RAFTER 

Web  connection;  buildings  at  right  angle;  unequal  pitches. 

For  the  sake  of  comparison,  we  can  so  modify  the  problem  shown  on  pp.  22  and  23  that  the  purlins  connect  to  the  web  of  the  valley  rafter  instead 
of  to  the  flange.  The  working  lines  and  the  column  connections  remain  the  same  and  it  is  unnecessary  to  show  the  latter  in  Fig.  18.  Given  b'  =  9'  1|", 
a'  =  6,  and  a"  =  4||-  Using  a  10"  I  25#  rafter  and  6"u8^t  purlins,  we  have  t  =  T55,  u'  =  u"  =  2,  and  r'  =  r"  =  5'  0".  The  required  values 
are  given  below. 

(For  necessary  computation  see  p.  47.) 


logarithm. 

Sine. 

Cosine. 

Tangent. 

A' 

o'  =  6 

9.65051 

9.95154 

9.69897 

A" 

•"-48 

9.56983 

9.96777 

9.60206 

C 

c=9f 

9.79567 

9.89258 

9.90309 

H 

A  =  3| 

9.97979 

9.49464 

X' 

-,  =llli 

9.84412 

0.01060 

X" 

*"-8A 

9  76344 

9.85249 

Number. 


Logarithm. 


6'  =  9'  If" 

0.95974 

e  =  4'  6H" 

0.65871 

d'  =  10'  2ft" 

6"=  11'  4f" 

1.05665 

d"=  12'  3|" 

m  =  14'  7ft" 

1  .  16407 

n  =  15'  3ft" 

?'"* 

P'  =  5F7^" 

0.74742 

s'  =  7'  6" 

Number. 

Logarithm 

w'  =  4H 

y\  =  &H 

9.74742 

?"=ft  4 

j"=  i 

a 

=  3'  8  9  " 

0.56983 

g' 

=  6'  2f  " 

W)' 

-2ft 

?/' 

=  3^ 

9.47292 

fc'  =  f 

z'  =  8i| 


CHATTER  III.     WEB  CONNECTION 


45 


;     LtM-TO 


U        :i...\ 


II.I.ISTRATIVE  PROBLEM 

CAMS  IV  (6)    VALLEY  IUnrR 

See  opposite  page. 


Fio.  18. 


46 


HIP  AND  VALLEY  RAFTERS 


ILLUSTRATIVE  PROBLEM 

CASE  IV  (a) 
Computation  of  values  given  on  p.  42. 


5"=  1.17337 

tan  A"  =  9. 69897 

e   =  0.87234 

V  =  1.07862 

tan  A'  =  9.79372 

V  =  1.07862 

cos  A'  =  9.92900 

d'  =  1.14962 

6"  =  1.17337 

cos  A"  =9. 95154 

d"=  1.22183 

tan  A"  =9. 69897 
tan  A'  =  9.79372 
tanC  =9.90525 

tan  A'  =9.79372 

sinC   =9.79698 

tan//   =9.59070 

V  =  1.07862 

sinC   =  9.79698 

m   =  1.28164 

cos#   =9.96930 

n  =  1.31234 


t/2  =8.19382 

sinC  =  9.79698 

i'=8.39684 


t/2 
tanC 


8.19382 
9.90525 


/=  8.28857 


r  = 

cos  A'  = 
tan  C  = 

Pr 

cos  C 
cos  H 


0.48755 
9.92900 
0.09475 
0.51130 
0.10827 
0.03070 


s'=  0.65027 


cos2  A' - 
tan//  = 
tan2  C  : 


9.85800 
9.59070 
0.18950 


w/=9.63820 


cos  A' 

cos  C       

cos X'=  9. 82073 


9.92900 
9.89173 


sin  A' =  9. 72272 

tan  C   =  9.90525 

y'  =  9.81747 

u'  =9.22185 

fc'  =9.03932 

tan  C   =  9 . 90525 

cos  A'  =  9.92900 

z'   =9.97625 

t/2   =  8.19382 
cosC   =9.89173 


{"=8.30209 

t/2   =8.19382 

tan  C   =  9 . 90525 

j"=8.09907 

r"=  1.00428 

cosA"=  9.95154 

tan  C   =  9.90525 

p"=  0.86107 

sinC   =0.20302 

cos//   =0.03070 

s"=  1.09479 


cos2A"=  9.90308 

tan2C   =  9.81050 

tan  H   =  9 . 59070 

w"=  9.30428 

cosA"=  9.95154 
sinC  =9.79698 
sinX"=  9.74852 

sin  A"  =9. 65051 

tan  C   =  9.90525 

y"=  9.55576 

u"=  9^22185 

fc"=8.77761 

cosA"=  9.95154 

tanC   =9.90525 

z"=9.85679 


Note.  —  No  distinction  is  made  between  logarithms  and  cologarithma  since  this  is  apparent  from  the  formulas. 


MIAPTEK  III.     WEB  CONNECTION 


47 


ILLUSTRATIVE  PROBLEM 

CASE  IV  (6) 
Computation  of  values  given  on  p.  44. 


b'  =0.95974 
t:m  A'  =  9.69897 

e  =0.65871 

tan  .1"=  J).(Ml-_'iM; 

6"=  1.05665 
1"=  9. 96777 
rf"  =  1.08888 

b'  =  0.  <>.V.  17 1 
1'  =9. 95154 
d'  =  1.00820 

tan  .1"=  9. 60206 
tan.l'  =9.69897 
taiiT  =9.90309 


tan.l' 
sin  (' 


9.69897 
9.79567 


tan//    =9.49464 

6'  =  0.95974 

siuC   =  9.79567 

m    =  1 . 16407 

DM  //    =  9.97979 


t/2 
sinC 


1/2 
tai.r 


8.11464 
9.79567 
8.31897 

8.11464 
9.90309 

s  -JII.V, 


^=0.69897 
cos  A'=  9. 95154 
tanC  -  0.09691 

p'=  0.74742 
cosC  =0.10742 
cos  H  =  0.02021 

s'=  0.87505 


oof  A' 

tan//  = 
tan'C  ° 


'.i  'Mi:;nx 
9.49464 
0. 19382 


w'-  9. 59154 

cos  A'  =9. 95154 
cosC  =  9.89258 
cos X'=  9. 84412 


MII  .r  - 1 

tanC   =9.90309 
y'  =9.74742 

u'  -  9.2-J l  v. 

k'  -  s.wi'.n.1; 

tanC   =9.90309 

owA'  -9.95154 

z'  -  9.95155 


8.11464 
9.89258 
8.22206 


1/2   -8.11464 
tanC  =9. 


n   =  1 . 18428 


j"-  8.01773 

r"-  0.69897 
cos  A"  =9. 96777 
tanC  =9.90309 

p"-  0.56983 
sinC  =  0.2(>»:« 
cos//  -0.02021 

«"-  6.79437 


co8M"-9.9:r.:.l 

tan'C  =9.80618 

tan//   -9.49464 

u>"-9.23636 

cos  A"  =9. 96777 
sinC  -  9.79567 
MM  X"-  9.76344 

sin  A"  =9. 56983 

tanC   -9.90309 

y"- 9. 47292 

u"-  9.22185 

*"-  8.69477 

008  A"  -9. 96777 

tanC  -9.90309 

z"- 9. 87086 


Nole.  —  No  distinction  is  made  between  logarithms  and  cologarithms  since  this  is  apparent  from  the  formulas. 


CHAPTER  IV 


NOTES  ON  OTHER   CASES 


ALTHOUGH  it  is  felt  that  the  majority  of  the  problems  in  hip  and  valley 
work  will  come  within  the  scope  of  the  notes  given  in  the  preceding  chap- 
ters, yet  conditions  will  occasionally  arise  which  demand  modifications, 
and  it  is  hoped  that,  with  the  aid  of  the  suggestions  which  follow,  the 
draftsman  will  be  able  to  adapt  one  of  the  first  six  cases  to  his  particular 
requirements. 

CHANNEL  PURLINS  WHOSE  FLANGES  FACE  THE  OTHER  WAY. 

1.  Flange  Connections.  —  The  connection  plate  will  generally  be  placed 
on  the  back  of  the  channel  web  as  before,  but  the  angle  of  bend  will  be 
obtuse  instead  of  acute,  and  the  details  for  the  hip  rafter  connections  will 
resemble  more  closely  those  for  the  valley  rafter  connections  of  Cases  I, 
II,  and  III.     In  like  manner  the  valley  rafter  connections  will  be  similar 
to  those  for  the  hip  rafters  of  Cases  I,  II,  and  III.     The  formulas  remain 
the  same. 

If  preferred,  the  bottom  flange  might  be  cut  away  to  allow  the  use  of 
the  same  details  as  before,  except  with  the  plate  placed  on  the  inner  face 
of  the  web.  In  this  case  the  working  lines  must  be  taken  to  the  back  of 
the  bent  plate  rather  than  to  the  back  of  the  channel.  The  former  method 
is  recommended. 

2.  Web  Connections.  —  The  connection  plate  should  be  placed  so  as  to 
avoid  a  sharp  bend,  in  order  to  facilitate  shop  work  and  erection.     This 
may  be  accomplished  in  two  ways,     (a)  By  cutting  the  purlin  short 
enough  to  permit  the  use  of  the  plate  shown  in  Cases  IV,  V,  and  VI,  the 
plate  being  bent  toward  the  channel  instead  of  away  from  it.     Care  should 
be  exercised  in  spacing  the  rivets  and  holes  to  give  sufficient  edge  distance 
and  driving  clearance.     (6)  By  using  a  plate  similar  to  the  above  and  nar- 
row enough  to  clear  the  flanges,  when  located  on  the  inner  face  of  the  web. 
The  latter  arrangement  is  preferable  unless  the  purlin  is  too  small  to  allow 
its  adoption.     In  either  event,  all  formulas  are  applicable,  provided  that 


the  working  lines  are  taken  to  the  back  of  the  bent  plate  rather  than  to 
the  back  of  channel.  It  will  not  be  necessary  to  cut  the  flanges  as  in 
Cases  IV,  V,  and  VI. 

I-BEAM  PURLINS. 

The  details  are  practically  the  same  as  for  channel  purlins  if  the  work- 
ing lines  are  taken  to  the  back  of  the  bent  plate  and  not  to  the  center  line 
of  web.  Part  of  the  flange  on  one  side  must  be  blocked  out  to  allow  for 
the  plate.  The  top  flange  may  have  to  be  cut  to  avoid  piercing  the  roof 
of  the  opposite  slope. 

Z-BAR  PURLINS. 

The  same  style  of  detail  may  be  used  to  connect  Z-bar  purlins  to  the 
hip  or  valley  rafter  as  for  channel  purlins  shown  in  Cases  I  to  VI  inclu- 
sive. The  flanges  must  be  cut  to  clear  the  rafter  and  to  avoid  piercing  the 
roof. 

ANGLE  PURLINS. 

Angle  purlins  may  be  attached  to  the  flange  of  the  rafter  by  connections 
similar  either  to  those  for  channel  purlins  or  to  those  shown  on  the  follow- 
ing pages  for  Tees.  The  web  connections  are  the  same  as  for  channels. 

TEE  PURLINS. 

The  connections  of  Tee  purlins  are  somewhat  peculiar  to  themselves 
although  similar  details  might  be  used  under  other  conditions.  The 
flanges  of  the  Tees  are  connected  to  the  flanges  of  the  hip  or  valley  rafters 
by  means  of  bent  plates.  It  is  deemed  unnecessary  to  illustrate  all  cases 
which  might  arise,  but  typical  details  of  both  hip  and  valley  connections 
are  shown  in  Figs.  19  and  20,  and  the  formulas  for  all  cases  appear  on 
pp.  49  and  50.  The  spacing  of  the  rivets  and  holes  is  apparent  without 
special  notes,  standard  gages  being  used  as  a  matter  of  course. 


48 


CHAPTER  IV.     NOTES  ON  OTHER  CASES 


I'. 


FORMl'l.VS    FOR  TEE  PURLIN  CONNECTIONS 
HIP  RAFTERS 

The  following  formulas  are  arranged  to  correspond  to  the  three  cases  of  hip  rafter  connections  for  channel  purlins  given  in  Chapter  II. 
The  following  values  are  either  given  or  else  obtained  from  the  proper  formulas  of  Cases  I,  II,  or  III:  — 

A',  a',  A",  a",  e,  b',  b",  d',  d",  C,  c,  H,  h,  m,  n,  r',  and  r". 
The  additional  data  required  may  be  found  by  means  of  the  formulas  listed  below. 

CASE  I  (a).  CASE  II  (a). 


Mil  // 
tllllC' 

tnnC 


(85)  x' 

2' 

(86)  x"=  tan  C  sin//. 
•(67)     2"  =  cos  A"  tan  C. 


t(87)  /"  = 

1(88)  v  = 

(89)  p'= 

(90)  s'  = 

(91)  p"  = 

(92)  s"  = 


cos// 

,  _  r'  cos  A'  cos  C  —  f 

sin  r 


_r 


\f*JJ    . 
*(59) 
(93)    ; 

•(73) 
i(R7^ 

taiif 
.      tanC 

cos  A' 
_,,          sin// 

tan  (L  -  C) 
,,         cos  A" 

tan  (L  -  C) 

x" 


(89) 
(90) 
(94)    p 
(95) 


YcosA'cosC-f 

>iuC 


CASE  III  (o). 
(96)    x   -sin//. 
(84)    2  -  cos  A  (sec  2"  in  Fig.  19). 

ft    _    Aft 

t(97)     /"-/'  +  -       -• 
z 

t  (98)    *  -  ' 
(99)    p'- 
(100)     «'- 


cos  45°  cos  H^  COB  U 


-('tan//. 


(101)    p"-r"cosA- 


r 

00045? 


„     r//cosA"co8(L-C)-/>' 
sin  (L  -  C) 


(io2)   »"-^^a 


-t"tan//. 


COS(L-C)C08// 


•  If  t  ho  value  of  :'  or  : "  U  greater  than  1 '  0"  the  bevel  should  be  reveraed  on  the  drawing  go  that  the  longer  side  beoomea  the  12"  base  and  the  shorter  aide  P  °*V' ' 

valur-i  am  obtainnl  ilirri-tly  fmin  I  ho  rologarithmg. 

t  /'  is  :is.-iitiii-.l  ci|ii.-il  ID  on,,  half  of  tho  flango  width  phu  a  distance  sufficient  to  allow  for  the  bend  in  the  plate.    /"  u  found  from  formula  (87)  or  (07).     If/"  is  too  Urge, 
it  may  bo  rvilurol  liy  increasing  /"  l>y  moans  of  filli-rs. 

t  D  is  the  amount  which  tho  hip  rafter  must  be  lowered  vertically. 


50 


HIP  AND  VALLEY  RAFTERS 


FORMULAS  FOR  TEE  PURLIN  CONNECTIONS 

VALLEY  RAFTERS 

The  following  formulas  are  arranged  to  correspond  to  the  three  cases  of  valley  rafter  connections  for  channel  purlins  given  in  Chapter  II. 
The  following  values  are  either  given  or  else  obtained  from  the  proper  formulas  of  Cases  I,  II,  or  III:  — 

A',  a',  A",  a",  e,  b',  b",  d',  d",  C,  c,  H,  h,  m,  n,  r',  and  r". 


The  additional  data  required  may  be  found  by  means  of  the  formulas  listed  below:  — 


CASE  I  (6). 

sinff 
tanC' 

tanC 


(85)  x'  = 

*(59)     z'  =  -  -77- 
cosA' 

(86)  x"=tanCsin#. 
*(67)     z"=  cos  A"  tan  C. 

x" 
.  f'x'  -  t' 


}(104)     v   = 
(105)    p'  = 


cosH 


(107) 
(108)     ^- 


cosC 


(85)  x'  = 

*(59)  z'  = 

(93)  x"  = 

*(73)  z"  = 

t(103)  /"  = 

}(104)  v  = 


CASE  II  (b). 

sinH 
tanC' 
tanC 
cos  A' 

sin  ff 
tan  (L  -  C)' 

cos  A" 
tan(L-C)' 
f'x'  -t'  + 1" 

x" 

f'x'  -t' 
cos  H 


(105) 
(106) 


p'-f'smC 


-  t'  tan  H. 


(110) 


cos  C cos  H 
,_r"cosA"cos(L-C)+/" 
sin  (L  -  C) 


cos  (L  —  C)  cos  « 


CASE  III  (6). 
(96)    a;   =  sinH. 

(84)     z   =  cos  A  (see  z"  in  Fig.  20). 

t(97)  r-r**-;?. 

}(98)     t;   =    "~   '  " 


(111)    p'=r'cosA 


sin45': 


(112)     s'  = 


P' 


cos  45°  cos 
(113)    p"=r"cosA  + 

(114) 


cos  H 


-  «'  tan  H. 


cos  45° 


p" f" 

sin  45°  cos  H      cos  H 


*  If  the  value  of  z'  or  z"  is  greater  than  1'  0"  the  bevel  should  be  reversed  on  the  drawing  so  that  the  longer  side  becomes  the  12"  base  and  the  shorter  side  —  or  —  •  These 
values  are  obtained  directly  from  the  cologarithms. 


fillers. 


~a  >u'    uvmuceu  UJTWUIJF  nuiii  MM  uuiuganuiiiiia. 

t  /'  is  assumed  approximately  equal  to  one  half  of  the  flange  width.    /"  is  found  from  formula  (103)  or  (97).     If  /"  is  too  large,  it  may  be  reduced  by  increasing  t'  by  means  of 


v  is  the  amount  which  the  valley  rafter  must  be  raised  vertically. 


CHAPTER  IV.     NOTES  ON  OTIII.l;  «   \>l.s 


51 


HIP  RAFTER 
Fia.  19. 


VALLEY  RAFTER 
Fia.  20. 


'  ana  •  nan  M  Vm  !••  M  M 


CHAPTER  V 


DERIVATION   OF  FORMULAS 


IN  this  chapter  are  given  the  derivations  of  all  formulas  which  involve 
anything  more  than  the  simple  trigonometric  relations  of  parts  of  a  right 
triangle,  or  the  substitution  of  other  values  in  formulas  already  deter- 
mined. These  derivations  are  by  no  means  essential  to  the  complete 
solution  of  a  problem,  but  are  inserted  rather  for  reference,  for  those  who 
have  difficulty  in  understanding  certain  features,  or  who  wish  to  investi- 
gate the  subject  more  thoroughly. 

Any  formulas,  other  than  those  given  here,  which  apply  to  the  connec- 

FORMULA  (7).     (Fig.  21.) 


But 

Substituting: 


tan  A' 


tan  C  =  ~ 


and    b"  = 


tan  A" 


tan  C  = 


tan  A" 
tan  A' ' 


FORMULA  (9).     (Fig.  21.) 


tan  H  =  —  • 
m 


But 
Substituting: 


e  =  b'  tan  A',    and    m  =  -: 


b' 


sin  C 
tan  H  =  tan  A'  sin  C. 


tions  of  the  purlins  in  the  steeper  roof  may  be  obtained  from  similar 
expressions  by  making  a  few  simple  changes  to  comply  with  the  new  con- 
ditions. 

Any  formulas  which  pertain  to  the  other  roof,  namely,  those  which  bear 
this  mark  ("),  are  similar  to  those  of  the  steeper  roof  and  may  be  derived 
from  them  by  changing  (')  to  (")  and  substituting  for  angle  C,  either  L-C 
or  90°-C.,  bearing  in  mind  the  relation  between  the  functions  of  an  angle 
and  the  functions  of  its  complement. 

FORMULA  (16).     (Fig.  21.) 
1-2 


But 

Substituting: 


tanC 

1-2  =  (r'-g')cosA'. 

,_  (/  -  q')  cos  A' 
P  tanC 

FORMULA  (17).     (Fig.  21.) 
1-3 


But 

Substituting: 


1-3  = 


cos  H 

2-3 

cos  C 

P' 


P' 


cos  C 


cos  C  cos  H 


52 


(  H.\ITi:n   V.     DKRI  \.\Tli  i\  OF  FORMULAS 


Fio. 21. 


Fio.22. 


FORMULA  (13).    (Fig.  22.) 

If  the  center  lino  of  the  top  flange  of  the  hip  rafter  were  placed  in  the  same  plane  as  the  top  of  the  main  rafters,  the  edge  of  the  flange  would  inter- 
fere with  the  purlin.  Thus,  if  the  bottom  of  the  purlin  cut  the  center  line  of  the  top  flange  of  the  hip  rafter  at  point  6,  and  the  top  of  the  main  rafter 
:it  jxiiiit  1,  the  vertical  plane  through  the  lx>ttom  of  the  purlin  would  cut  the  edge  of  the  hip  rafter  flange  at  point  7.  Since  the  web  of  the  hip  rafter 
is  vertical,  and  the  llange  normal  to  the  web,  points  7  and  8  are  at  the  same  elevation,  as  shown  at  points  2  and  3  respectively.  Therefore,  the  hip 
rafter  should  be  lowered  vertically  a  distance  1-2  plus  any  desired  clearance,  as  \"  or  ft". 


1-2  =  (2-3)  tan  A'. 
But  2-3  =  (7-8)  cos  C,    and     7-8 

Substituting :  1-2  =  { tan  A '  cos  C, 


and  r  =     tan  4'  cosC  +  J". 


./. 
2 


Note.  —  In  tho  derivation  of  the  following  formulas,  the  clearance  }"  is  omittwl  for  simplicity,  since  the  angle*  remain  unchanged, 


54  HIP  AND  VALLEY  RAFTERS 


FORMULA  (18).     (Fig.  23.) 

From  the  derivation  of  Formula  (13)  it  may  be  seen  that  if  the  purlin  should  rest  on  the  flange  of  the  hip  rafter  at  point  7  in  the  plan,  or  11  in 
the  plane  of  the  flange  of  the  hip  rafter,  the  center  line  of  the  top  flange  of  the  rafter  at  point  6  would  be  a  vertical  distance  1-2  below  the  purlin. 
The  plane  of  the  purlin  web  would  cut  this  center  line  at  point  4,  which  corresponds  to  point  9  in  the  plan  and  13  in  the  plane  of  the  flange  of  the 
rafter.  The  line  11-13  is,  therefore,  the  line  of  intersection  of  the  plane  of  the  purlin  web  and  the  plane  of  the  top  flange  of  the  hip  rafter,  and 
hence  the  line  of  bend  of  the  connection  plate. 

Let  w'  represent  the  tangent  of  the  angle  which  this  line  makes  with  the  center  line  of  the  flange;  then 

,      12-13 

w  = 


But  11-12  = 


11-12 
8-9  3-5 


cos  H      sin  C  cos  H 
where     .  3-5  =  (3-4)  cos  A'  =  (2-3)  cos2  A', 

and  2-3  =  (7-8)  cos  C; 

(7-8)  cos  C  cos2  A' 

therefi.r  11-12  = 4—        -^ 

sm  C  cos  H 

Also  12-13  =  I  =  7-8. 

Substituting  these  values  above: 

7-8 


(7-8)  cos2  A' 
tan  C  cos  H 

tan  C  cos  H 
cos2  A' 


<  HAITKR  v.     ni:i:i\.\Ti't\  «n-  IOUMU.AS 


B5 


l  ...  •_'.; 


56  HIP  AND  VALLEY  RAFTERS 

FORMULA  (19).     (Fig.  23.) 

The  intersection  of  the  plane  of  the  purlin  web  and  the  plane  of  the  top  flange  of  the  hip  rafter  is  the  line  11-13  or  20-21.  The  line  7-6,  11-15 
or  20-22  is  the  bottom  line  of  the  purlin  web.  If  a  plane  is  passed  through  the  point  13,  or  21,  perpendicular  to  the  line  of  intersection  it  will  cut 
the  plane  of  the  purlin  web  in  a  line  21-22,  or  18-19,  and  the  projection  of  this  line  upon  the  plane  of  the  top  flange  of  the  rafter  will  be  13-14 
or  17-19.  The  angle  X',  or  the  complement  of  the  angle  between  the  two  planes,  is  shown  in  its  true  size  at  17-18-19  and 

.    y,  =  17-19  =  13-14 
"  18-19  "21-22" 

But  13-14  =  (11-13)  tan  13-11-14  =  (11-13)  tan  (W  -  K), 

and  21-22  =  (20-21)  tan  21-20-22  =  (11-13)  y'. 

.    v,     tan  (IP' -X)          tanTF'-tan-K 
Therefore  sm.X'=-     - — —     -  =  .         — =7- — ^T"'' 

,     tan  C  cos  H  „      15-16  ,_  1C      /      _  0 

But  tan  W  =  w'  =  -    — 2  .,         and    tan  K  =  -     ^  >    where     15-16  ™  5  ™  7-8, 

COS    A.  11    lo  « 

„      (7-8)  cos  H                            „.      tanC 
and  11-16  =  (6-8)  cos  H  =  „ — »    whence    tan  K  = jj  • 

tan  C  cos  H      tan  C 

.    „,         cos2  A'          cosH       tan  C  (cos2  H  —  cos2  A') 
Substituting  these  values:  sin  A  =  — -f — ^- 


cos2  H  -  cos2  A'=  cos2  tf  [1  -  cos2  A'  (1  +  tan2#)] 

=  cos2  H[l  -  cos2  A'  -  cos2  A'  tan2  A'  sin2  C]        See  Formula  (9). 
=  cos2  H  [sin2  A'  -  sin2  A'  sin2  C] 
=  cos2.ff  sin2  4'  cos2  C. 


cos2  A'  +  tan2  C  =  [cos2  C  +  sin2  C  (1  +  tan2  A')] 

os2  c  +  sin2  c  +  sin2 


cos2  A'  ,, 

Wc[1+tan//] 
cos"  A' 


cos2  C  cos2  H 

From  the  following  derivation  (p.  57): 

,  _  sin  A '  sin  C  cos  C  cos2  H 
cos2  A' 

Substituting  these  three  values  and  reducing: 

sin  X'  =  sin  A'  cos2  C  cos  H. 


CHAPTER  V.     1)1  i;l\  ATION  OF  FORMULAS 


57 


MULA  (21).     (Fig.  23.) 

If  y'  represent  the  tangent  of  the  angle  in  the  plane  of  the  purlin  web  between  the  line  of  bend  7-9  and  a  horizontal  line  7-0,  then 

2-4 


But 


where 


Substituting: 


But 


Substituting: 


7-10 
2-4  -  (1-2)  COB  A'  =  ^ sin  A' cos  C    and    7-10  -  (7-6)  -  (10-6), 

,_-  2  9-10        5-2        (2-1)  sin  A' 

7-6  =  -i — 77      and     10-6  =  r  -75  ™  ; — 7,  "  ~  — 

sin  C  tanC      tanC 


tanC 


.  i-in  A'cosC 


/         /sin*  A 'cos'C' 
2  sin  C  2  sin  C 

,_  sin  A' sin  C  cos  C 
y  ~  1  -sin* A' cos* C 

I  -  sin*  A'  cos*  C  -  1  -  sin*  A'  +  sin*  A'  sin*  C 
=  cos*  A' +  sin*  A' sin*  C 
-  cos*  A'  (1  +  tan*  A'  sin*  C) 
=  cos*  A' (1+ tan*//), 

cos*A' 
"  cos*//' 

sin  A '  sin  C  cos  C  cos*// 


cos'A' 


If  we  multiply  Formula  (18)  by  Formula  (19)  we  have: 

tan  C  cos  H  . 


tc'sinA" 


Therefore 


sin  A' cos*  C  cos  H 

y'=u>'sinA". 


sin  A'  sin  C  cos  C  cos*  H 
cos*  A' 


58 


HIP  AND  VALLEY  RAFTERS 


FIQ.  24. 


FIG.  25. 


tanC 


FORMULA  (30).     (Fig.  24.) 
1-2  b' 


(3-4) +  (2-3)        6"  b' 

.T; 


sinL      tanL 


V  sin  L 
+  b'cosL 


tanC  = 


FORMULA  (31).     (Fig.  25.) 
1-2  b' 


(3-4)  -  (3-2)        b" V_ 

sin  L      tan  L 

6'sinL 


c"-6'cosL 


CHAPTER  V.     DERIVATION  OF  FORMULAS 


59 


Fia.  26. 
FORMULA  (54).     (Fig.  26.) 

The  line  2^-24  is  the  intersection  of  the  plane  of  the  purlin  web  and  the  vertical  plane  through  the  center  line  of  the  main  rafter,  and  26-29  the 
intersection  of  the  plane  of  the  purlin  web  and  the  vertical  plane  through  the  center  line  of  the  hip  rafter.  Let  us  consider,  for  illustration,  that 
part  of  the  purlin  web  whose  vertical  projection  is  measured  by  the  distance  »'  =  23-25  =  26-27.  If  the  line  26-28  is  drawn  perpendicular  to  the 
top  of  the  hip  rafter,  the  angle  27-26-28  =  H.  Let  the  angle  27-26-29,  which  the  line  of  intersection  26-29  makes  with  the  vertical,  be  represented 
by  M.  Then  w'  =  tan  28-26-29  =  the  tangent  of  the  complement  of  the  angle  between  the  line  of  bend  and  the  top  flange  of  the  hip  rafter. 


But 


Substituting: 


tan  Af 


tc   — 
27-29 

ian  .fio-zo-zy  =  ian  \ai  —  n  )  =»  r-^ 
24-25 

tan  M  tan  // 
tan// 

-in  i  '      c'tanA 

>     and     tan  A' 
tantf(l  -sin'C) 

26-27 

in' 

v'          v'  sin  C 
tann       .  .     „ 

sinC 
»«C 

.  ,  ,:      ian  ti 
sin2  C 

tnn'tf 
^sitfC 
tan//  cos*  C 

s^C  +  tan1^ 

tan  //  i-i 

sin*  C  +  tan*  A'  sin1  C      sin*  C  (1  + 

tan'  A')' 

tanM 


tan// 


cos*  A' tan  H 
tan*C 


60 


HIP  AND  VALLEY  RAFTERS 


FIG.  27. 

FORMULA  (55).     (Fig.  27.) 

The  line  26-29  is  the  intersection  of  the  plane  of  the  purlin  web  and  that  of  the  hip  rafter  web,  and  M  is  the  angle  which  it  makes  with  the 
vertical.  C  ( =  33-34-35)  is  the  horizontal  projection  of  the-angle  between  these  two  planes,  but  the  true  angle  is  measured  at  right  angles  to  the  line 
of  intersection.  If  the  plane  containing  this  angle  is  revolved  into  the  horizontal  about  33-35,  an  axis  normal  to  the  rafter  wet),  the  point  30  will  fall 
at  31  in  the  elevation,  or  32  in  the  plan,  and  the  required  angle  X'  between  the  two  planes  will  appear  in  its  true  size  at  33-32-35. 

33-35      (34-35)  tan  C  _  (27-29)  tan  C 


tan  X'  = 


32-35  27-30 

v'  tan  M  tan  C      tan  C 


From  the  derivation  of  Formula  (54),  p.  59,  we  have: 

tanM  = 


v'  sin  M 


cosM 


tan  A' 
sinC 


whence    cosM  = 


v'  sin  M 


1 


Substituting  : 


But 


tanX'  =  tanCVl  + 

1 


I         tin2  A'  I 

y  1  +  ^nj  =  \ 


tan2  C  + 


tan2  A' 
cos2  C  ' 


cos  X'  =  - 


tan- A' 
sin2C 


cosC 


Vl+tan2X' 
cosX'  =  cos  A'  cos  C. 


s/ 


l+tan2C 


tan2  A!      Vl+tan2A' 
cos2C 


CIIMTKK  V.      DERIVATION   (H- 


61 


fc 


I'M;.  28. 

FORMULA  (57).     (Fig.  28.) 

The  horizontal  projection  of  the  line  of  intersection  of  the  web  of  a 
purlin  of  depth  u  and  the  rafter  web  is  represented  by  the  line  35-37. 
The  bottom  flange  of  the  purlin  must  be  a  distance  35-36  shorter  than  the 
top  flange.  The  cut  is  shown  in  the  plane  of  the  purlin  web  by  the  angle 
whose  tangent  is  y'.  36-37 

,     35-36      tanC       24-25 


tanC 
u 

Mil  .1  ' 


u  tan  C      u  tan  C 


FORMULA  (59).     (Fig.  28.) 

The  purlin  flanges  are  cut  at  an  angle  C  measured  horizontally,  and 
this  heroines  the  angle  whose  tangent  is  z'  when  measured  in  the  plane  of 
the  purlin  flange.  36-37 

,  =  39-40  =  23-34      cosT' 

"  :$s  ;{..i  "  ::;,-:<!; 


tanC 

MM  .1 ' 


tanC' 


Fiu.  29. 


FORMULA  (85).    (Fig.  29.) 

For  simplicity,  let  us  consider  that  the  edge  of  the  top  flange  of  the  hip 
rafter  lies  in  the  plane  of  the  roof,  and  is  therefore  the  line  of  lx>nd.  If  a 
plane  is  passed  perpendicular  to  this  line  of  hend  it  will  cut  the  top  flange 
in  the  line  42-44  and  the  roof  plane  in  the  line  42-45.  The  angle  of  bend, 
or  the  angle  between  the  planes  of  the  top  flange  and  the  roof,  is  shown 
in  the  plan  at  44-42-45,  and  in  its  true  size  at  50-49-51.  Let  x'  represent 
the  tangent  of  this  angle;  then, 


tan  50-49-51  - 


.Mi   :,1 
I'.'    .Ml' 


But 
and 

Substituting, 


50-51  -  47-48  -  (46-48) sin//  =  (41-13) sin H, 
49-50  =  /'  =  43-45  =  (41^3)  tan  C. 

j      (41-43)  sing      sing 
"  (41-43)  tan  C"  tan  C* 


CHAPTER  VI 


GRAPHIC  METHOD   OF  DETERMINING  ANGLES 


THE  angles  used  in  the  various  purlin  connections  of  the  preceding 
chapters  may  be  determined  graphically  if  desired,  but  the  linear  dimensions 
should  be  calculated  in  every  case,  for  the  ordinary  precision  of  the  graphic 
method  is  not  sufficient  to  insure  the  best  results.  In  fact,  it  is  believed 
that  the  complete  algebraic  method  is  quicker  as  well  as  more  satisfactory, 
and  the  graphic  method  is  recommended  merely  as  a  check. 

Since  the  angles  are  the  same  for  either  hip  or  valley  work,  the  drawings 
of  this  chapter  show  only  valley  rafters.  For  the  sake  of  clearness,  the 
determination  of  each  angle  is  illustrated  separately,  as  well  as  in  con- 
junction with  the  other  angles  occurring  in  the  same  case.  The  drawings 
show  only  those  lines  which  are  necessary  in  the  determination  of  the 
different  angles,  and  no  attempt  has  been  made  to  show  the  complete 
projections  of  Descriptive  Geometry.  The  accompanying  descriptions, 
therefore,  cannot  be  given  in  minute  detail,  but  it  seems  better  not  to 


complicate  the  drawings  in  order  to  simplify  the  analyses.  These  descrip- 
tions refer  to  the  angles  used  in  the  connections  of  the  purlins  in  the 
steeper  slope;  they  may  be  made  to  apply  also  to  the  corresponding  angles 
of  the  other  slope  by  substituting  (")  for  ('),  and  in  some  cases  taking  the 
complement  of  the  angle  thus  found.  The  actual  value  of  the  angle  is 
seldom  necessary,  but  the  slope,  or  tangent  of  the  angle  for  a  base  of  one 
foot,  may  be  scaled  directly  from  the  drawing. 

There  are  four  planes  of  projection,  one  horizontal  K'GK",  and  the 
other  three  vertical  through  the  webs  of  the  main  and  valley  rafters, 
D'B'E',  D"B"E"  and  MGN,  respectively.  The  angles  A',  A",  C  and  H 
are  plotted  from  the  given  data,  and  the  line  P'GP",  drawn  perpendicular 
to  GM,  is  the  H  trace  (horizontal  trace)  of  the  plane  of  the  top  flange  of 
the  valley  rafter. 


FLANGE  CONNECTION 


ANGLE  W  (Fig.  30),  in  the  plane  of  the  top  flange  of  the  valley  rafter, 
between  the  line  of  bend  and  the  center  line  of  the  rafter. 

The  plane  of  the  purlin  web  cuts  the  center  line  of  the  rafter  at  a'  in  the 
plan,  or  b'  in  the  elevation.  It  also  cuts  the  H  trace  of  the  plane  of  the 
top  flange  of  the  rafter  at  P'.  Thus  we  have  two  points  in  the  line  of 
intersection  of  the  two  planes.  If  the  plane  of  the  top  flange  of  the  rafter 
is  revolved  into  H,  about  its  H  trace,  GP',  b'  will  fall  at  c'  and  the  line 
of  intersection  at  c'P'.  The  angle  P'c'G  between  this  line  and  the  center 
line  of  the  rafter  will  be  the  required  angle  W. 

ANGLE  X'  (Fig.  31),  the  complement  of  the  angle  between  the  plane  of 
the  top  flange  of  the  valley  rafter,  and  the  plane  of  the  purlin  web,  or 
the  angle  of  bend. 

If  the  line  of  intersection  of  the  two  planes  be  revolved  into  H  about 
its  H  projection,  a'P',  it  will  fall  at  d'P'.  A  plane  normal  to  the  line  of 


intersection  at  any  point  g'  will  cut  the  H  trace  of  the  plane  of  the  top 
flange  of  the  rafter  at  /',  and  the  H  trace  of  the  plane  of  the  purlin  web 
at  j'.  If  this  normal  plane  is  revolved  into  H  about  its  H  trace,  f'j',  g' 
will  fall  at  h',  and  the  obtuse  angle  between  the  plane  of  the  top  flange 
of  the  rafter  and  the  plane  of  the  purlin  web  will  show  in  its  true  size  at 
f'h'j'.  If  a  line  be  drawn  perpendicular  to  f'h'  at  h',  the  angle  between 
this  line  and  the  line  h'j'  will  be  the  required  angle  X',  or  the  complement 
of  the  acute  angle  between  the  two  planes. 

ANGLE  Y'  (Fig.  32),  in  the  plane  of  the  purlin  web,  between  the  line  of 
bend  and  a  horizontal  line. 

If  the  plane  of  the  purlin  web  is  revolved  into  H  about  its  H  trace,  MP', 
the  line  of  intersection  of  the  plane  of  the  purlin  web  and  the  plane  of  the 
top  flange  of  the  rafter,  or  the  line  of  bend,  will  fall  at  m'P',  and  the 
angle  m'P'M  will  be  the  required  angle  Y'. 


62 


CHAITKK  VI.     ORAl'llH     MITHOD 


63 


Wand 


Fio.  30. 


Y'  and  Y" 


Fia.  32. 


FLANGE  CONNECTION. 
For  combined  layout  see  Fig.  33  or  Fig.  34. 


64 


HIP  AND  VALLEY  RAFTERS 


CASE   I 


CASE   H 


FIG.  33. 


FLANGE  CONNECTION. 
For  separate  layouts  see  Figs.  30,  31  and  32. 


Fio.  34. 


M       (.KM'IIH     MITIInD 


W'andW 
FlO.  35. 


\\  i.  n  CONNECTION. 
For  combined  layout  sec  Fig.  39  or  Fig.  40. 


WEB  CONNECTION. 


AM-.LE  II"'  (Fig.  35),  in  the  plane  of  the  web  of  the  valley  rafter,  the 
complement  of  the  angle  between  the  line  of  bend  and  the  top  flange  of 
the  rafter. 

The  phino  of  the  purlin  web  cuts  the  center  line  of  the  top  flange  of 
the  rafter  at  n'  in  the  plan,  or  6'  in  the  elevation.  It  also  cuts  the  //  trace 
of  the  plane  of  the  rafter  \ve!>  at  M.  Ml/  is,  therefore,  the  line  of  inter- 
section of  the  two  planes,  or  the  line  of  bend,  and  the  angle  between  this 
line  and  a  line  normal  to  the  top  flange  is  the  required  angle  W . 

AM.I.K  A  '  Fig.  36),  Ix'tween  the  plum-  of  the  purlin  web  and  the  plane 
of  the  web  of  the  valley  rafter,  or  the  angle  of  bend. 

If  a  plane  be  passed  through  a'  perpendicular  to  the  line  of  intersection 
of  the  two  planes  at  n',  it  will  cut  the  //  trace  of  the  plane  of  the  rafter 


web  at  a',  and  the  H  trace  of  the  plane  of  the  purlin  web  at  p'.  If  this 
normal  plane  is  revolved  into  H  about  its  H  trace,  a'p',  n'  will  fall  at  o', 
and  the  required  angle  X'  between  the  two  planes  will  be  shown  in  its 
true  size  at  a'o'p'. 

ANGLE  Y'  (Fig.  37),  in  the  plane  of  the  purlin  web,  the  complement 
of  the  angle  between  the  line  of  bend  and  a  horizontal  line. 

If  the  plane  of  the  purlin  web  is  revolved  into  //  about  its  H  trace 
MK",  the  line  of  intersection  of  the  plane  of  the  purlin  web  and  the 
plane  of  the  rafter  web,  or  the  line  of  bend,  will  fall  at  Mm'  and  the 
angle  m'MK"  will  l>e  the  angle  which  it  makes  with  a  horizontal  line. 
The  complement  of  this  angle  is  shown  at  Mm'a'  and  is  the  required 
angle  1". 


66 


HIP  AND  VALLEY  RAFTERS 


B" 


Z'andZ" 
FIG.  38. 


Case  IV 
FIG.  39. 

WEB  CONNECTION. 
For  separate  layouts  see  Figs.  35,  36,  37  and  38. 


Case  V 
FIG.  40. 


WEB  CONNECTION. 

ANGLE  Z'  (Fig.  38),  in  the  plane  of  the  roof,  between  the  center  line  of     horizontal  line  MK",  the  center  line  -of  the  rafter  will  fall  at  Mq',  and 
the  valley  rafter  and  a  horizontal  line.  the  angle  K"Mq'  will  be  the  required  angle  Z'. 

If  the  plane  of  the  roof  is  revolved  into  a  horizontal  position  about  the 


(  1IAITKH  VI.     GRAPHIC   Ml  Tl|o|i 


1,7 


X  and  X" 


Fia.  41. 


CASE  I 


Fio.  42. 


TEE  PtRiJN  CONNECTION. 
•For  doparate  layout  for  angles  Z'  and  Z"  see  Fig.  38  (Web  Connection). 


TEE  PURLIN  CONNECTION. 
ANCI.K  .V  (Fig.  41),  between  the  plane  of  the  roof  and  the  plane  of  the     its  H  trace  KV,  s'  will  fall  at  (',  and  the  line  of  intersection  bet\v<  m 


top  flange  of  the  valley  rafter,  or  the  angle  of  bend. 


the  normal   plane  and  the  plane  of  the  roof  will  fall  at  K'l'.    The 


A  plane  through  any  point  K'  in  the  H  trace  of  the  roof  plane,  normal     angle  t'K'r1  between  this  line  and  a  horizontal  line  will  IK-  the  required 
to  the  tlai\i;r  of  the  valley  rafter  would  cut  the  flange  in  a  horizontal  line     angle  X'. 
through  the  point  s'.    If  this  normal  plane  is  revolved  into  H  about        ANGLE  Z'  (Fig.  38),  the  same  as  for  web  connection,  p.  66. 


68 


HIP  AND  VALLEY  RAFTERS 


Pitches  J  and  30°- 

Angle. 

Slope. 

Logarithm. 

Flange 
connections. 

5»  -JO* 

w"=8^ 

x'  =31 
x"  =  2f 

2/'=4 
*"-3A 

Sine. 

Cosine. 

Tangent. 

A' 
A" 

a'  =8 
a"  =  6{f 

9.74406 
9.69897 

9.92015 
9.93753 

9.82391 
9.76144 

Web 

connections. 

w'=4H 
w"=2ft 

i=»« 

z"=8i 

2/'  =7H  log  y'  =9.80653 
y"  =  5t\  log  y"  =  9.  63650 

i=m 

2"  =  9 

C 
II 

c  =  10| 
h  =  5l 

9.81601 

9.87848 
9.96214 

9.93753 
9.63992 

Tee 

connections. 

x'  =  5  A 
x"  =  4ft 

|  .... 

2"  =  9 

Pitches  3  and  ^  • 

Angle. 

Slope. 

Logarithm. 

Flange 
connections. 

i=11^ 

u>"  =  7f 

x'  =4A 

x"  =  \ti 

2/=4 
j/"=2| 

Sine. 

Cosine. 

Tangent. 

A' 
A" 

a'  =8 
a"  =  6 

9.74406 

9.65051 

9.92015 
9.95154 

9.82391 

9.69897 

Web 
connections. 

w'  =5tf 
w"  =  2fs 

?  =1°tt 
x"  =  7| 

j/'  =8|log!/'  =9.86900 
j,"  =  4    logy"  =  9.  52557 

2'  =1011 
«"-8* 

C 
H 

c  =  9 
A  =  4H 

9.77815 

9.90309 
9.96777 

9.87506 
9.60206 

Tee 

connections. 

*'  =5« 
^"=3ft 

2'  =io« 
•"-8* 

P 

tches  IT  and 

t- 

Logarithm. 

Flange 

to'  =9H 

x'  =5 

2/'=3M 

Sine. 

Cosine. 

Tangent. 

connections. 

«>"  =  6ft 

*"-« 

2/"  =  2A 

4' 
4" 

a'  =8 
a"=4H 

9.74406 
9.56983 

9.92015 
9.96777 

9.82391 
9.60206 

Web 

connections. 

«>'=7H 
w"=H 

x'  =HH 
x"  =  6| 

y'  =  Hi  logy'  =9.96591 
y"=2ft  log  j/"  =  9.  34798 

2'  =8| 

2"  =  6M 

Q 

c-7A 

9  71138 

9.93323 

9.77815 

x'  =6J 

2'  -8| 

If 

A-4i 

9.97585 

9.53529 

Tee 

connections. 

r"  =  2A 

z"  =  6ti 

TAIU.KS 


Pi 

U.U.  4  and 

\. 

An«le. 

-...),- 

Ixumrilhm. 

i    ,,, 

u>'-8A 

i'  -5| 

*'-3A 

COMM. 

Tucwt. 

w"-5| 

/"-I 

V"-1A 

A' 

A" 

a'  -8 
a"-4 

0.74406 
9.50000 

0.02015 
0.07712 

0.82301 
0.52288 

Web 

connection*. 

»'  -9« 
«>"-« 

z'  -101 
z"-5| 

i  -10JJ  logy'  -0.04509 
»"-U     logv"-9.  19807 

«'-7A 
«"-5H 

C 

c-6 

0.65051 

0.05154 

0.60807 

z'  -61 

i'  -7A 

TM 

11 

A-3A 

0.08151 

0.47442 

z"-Hl 

*"-5li 

Pitch*  30°  «d  1  . 

Antic. 

Slop.. 

LocAnthm. 

i  ;  mm 

.T-IH 

ir 

ic"-8I 

x'-3A 
x"-2A 

V'-3A 

y"-2I 

BH 

COKM,. 

Tu^Bt. 

.1' 
A" 

°'-«H 

a"-6 

0.69807 
0.65051 

0.03753 
0.95154 

0.76144 
0.69807 

W»b 

on«n»itioM 

«'  -4A 

u>"-2| 

p-IO| 
x"-8H 

y'  -6f|log/  -9.70144 
y"-4|    log  y"-9.  58804 

*'  -12 
«"-OA 

C 

II 

c-lOj 
A-4A 

0.81601 

9.87848 
9.07100 

0.03753 
0.57745 

Tea 

.  MM  '•••'•• 

x'-4i 
x"-3H 

s'  -12 

*"-OA 

PitcbM30°udi- 

Anile. 

••i 

Lacarilbm. 

1  '   Ml 

u>'  -  10J 
tr"-7A 

*'-4A 
*"-l| 

»'-3| 
»"-2A 

•Mb 

COM*. 

Taocmt. 

A' 

.1 

a'-6« 
a"-4« 

9.60807 
9.56983 

0.03753 
0.96777 

9.76144 
9.60206 

M 

OOUMCtioU. 

w'  -6A 

U>"-1| 

i'-HH 
z"-7J 

y'  -Stilogy'  -9.85835 
y"-3A  log  v"-9.  41045 

«'  -0| 

*"-7| 

C 

// 

c-8A 
*-3}| 

9.75540 

0.01487 
0  97771 

9.84062 
9.51693 

Tee 

.  01  :..•  MM 

«'-5A 
i"-2| 

z'  -Of 
x"-7l 

70 


HIP  AND  VALLEY  RAFTERS 


Pitches  30°  and  6- 

Angle. 

Slope. 

Logarithm. 

Flange 
connections. 

w'  =  81 
w"-«| 

z'=4f 
z"=H 

y'=3A 
>r"-«i 

Sine. 

Cosine. 

Tangent. 

A' 
A" 

a'=6M 
a"  =  4 

9.69897 
9.50000 

9.93753 
9.97712 

9.76144 

9.52288 

Web 

connections. 

W  =7« 
w"=lA 

*'  =10  A 

*"  =  6A 

y'  =10f  logy'  =9.93753 
j/"  =  2A  log  j/"  =  9.  26144 

2'  =8 

*"-8A 

C 
H 

c=m 

A  =  3A 

9.69897 

9.93753 
9.98262 

^9.76144 
9.46041 

Tee 

connections. 

x'  =51 

*"-m 

2'  =8 
2"=6ft 

Pitches  1  and  i- 

Angle. 

Slope. 

Logarithm. 

Flange 
connections. 

W  =11  A 

w"=m 

x'  =3^ 
*"-!« 

y>  =3 

2/"  =  2ft 

Sine. 

Cosine. 

Tangent. 

A' 
A" 

o'  =6 
a"  =  4H 

9.65051 
9.56983 

9.95154 
9.96777 

9.69897 
9.60206 

Web 

connections. 

w'  =4H 
w"  =  2& 

p-lltt 

z"  =  8ft 

y'  =6ttlog2/'  =9.74742 
y"  =  3A  log  y"  =  9.  47292 

2'  =10J 
«"-8« 

C 
H 

c=9f 
h  =  3l 

9.79567 

9.89258 
9.97979 

9.90309 
9.49464 

Tee 
connections. 

x'  =4^ 
i"=2J 

2'  =10^ 

2"  =  8M 

Pitches  i  and  i- 

Angle. 

Slope. 

Logarithm. 

Flange 
connections. 

w'  =9| 

w"  =  "\ 

x'  =3| 
*"=H 

J/'  =  2I 
/'-!« 

Sine. 

Cosine. 

Tangent. 

A' 
A" 

a'  =6 
o"  =  4 

9.65051 

9.50000 

9.95154 
9.97712 

9.69897 
9.52288 

Web 

connections. 

w'  =6 
w"-lft 

*'  =io| 

*"  =  7A 

y'  =8  A  logy'  =9.82660 
y"  =  2i    log  y"  =  9.  32391 

2'=8B 
2"  =  7A 

C 
H 

c  =  8 
A  =  3,^ 

9.74406 

9.92015 
9.98391 

9.82391 
9.44303 

Tee 
connections. 

*'=4« 
x"=2J 

2'  =8« 

2"  =  7A 

TABLES 


71 


PitehMiud  4- 

Antle. 

-,•;.- 

Logarithm. 

1     ,,.. 

io'  -nj 

w"-9A 

x'-2| 
i"-lj 

*'-2| 
»"-!« 

•• 

CoriM. 

TUCMI. 

A' 

A" 

o'-4H 
a"-4 

0.50663 
9.  60000 

9.96777 
9.97712 

9.60206 
9.52288 

M 

BOMMrtlOM. 

w'-3tt 
u,"-lH 

I'-llH 
x"-9A 

y'  -51    logy'  -9.64901 
¥"-  3  A  logy"  -9.  42082 

i'  -10] 
•"-9J 

C 

11 

c-10 

*-3A 

9.80631 

9.88549 
9.98621 

9.92082 
9.40837 

Tm 

MM  •..  •  • 

x'-3A 
z"-2} 

t'  -101 
t"-9J 

I 

qul  pitch.. 

i- 

Logarithm. 

1  ,  •,,. 

\-.  i 

Slap*. 

•M 

Caria». 

TucMt. 

.-..,,„-,,,. 

io-9A 

z-3J 

V-3H 

A 

0-8 

9.74406 

9.92015 

9.82391 

W*b 

tc-3fj 

i-8l 

y-6|iloRV-9.  74406 

(-10 

C-45" 

c-12 

9  84949 

9  84949 

TM 

z-51 

f-10 

// 

A  -5H 

9  95642 

9  67340 

Kqiml  pilehw  30°- 

Angle. 

Slopr. 

Locu-ithm. 

i   mm 

COOMCtWO.. 

v-91 

z-2I 

V-3A 

.-:,,, 

Cora*. 

Tunat. 

A 

a  =  6« 

9.69897 

9.93753 

9.76144 

W*b 

COOB«CtkMW. 

U7-3H 

z-9A 

V-6  log  y-9.  69897 

(-101 

C-45° 

// 

c-12 
A-4I 

9.84949 

9.84949 
9.96653 

TM 

*-4A 

(-10| 

9.61093 

72 


HIP  AND  VALLEY  RAFTERS 


B 

qual  pitches 

I- 

Logarithm. 

Flange   ' 

Sine. 

Cosine. 

Tangent. 

connections. 

w  —  10f5 

x-2fg 

j/-3 

A 

0  =  6 

9.65051 

9.95154 

9.69897 

Web 
connections. 

u)  =  3| 

z  =  9H 

2/  =  5|logi/  =  9.65051 

z  =  10| 

C-45° 

c  =  12 

9.84949 

9  84949 

Tee' 

z  =  4 

z  =  10f 

H 

h=4l 

9.97442 

9.54846 

Equal  pitches  5  • 

Angle. 

Slope. 

Logarithm. 

Flange 
connections. 

s  =  2^ 

P-3| 

Sine. 

Cosine. 

Tangent. 

A 

o=4H 

9.56983 

9.96777 

9.60206 

Web 

connections. 

W-SH 

x  =  10A 

!/  =  4A  log  y=9.  56983 

f-lli 

C  =  45° 
H 

c  =  12 
A  =  3f 

9.84949 

9.84949 
9.98328 

Tee 

connections. 

*»3t 

f-lH 

9.45155 

Equal  pitches  ff  • 

Angle. 

Slope. 

Logarithm. 

Flange 
connections. 

w  =  ll| 

|f-2 

Sine. 

Cosine. 

Tangent. 

x-\l 

A 

0  =  4 

9.50000 

9.97712 

9.52288 

Web 

connections. 

w  =  2& 

x  =  10J 

y=Sn  log  y  =  9.  50000 

2  =  111 

C  =  45° 
H 

c  =  12 

A=2if 

9.84949 

9.84949 
9.98826 

Tee 

connections. 

x  =  2J 

2  =  111 

9.37237 

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